10th grade lesson plan packet - great hearts irving ... · 5/10/2020 · m on d ay, m ay 11...
TRANSCRIPT
10th Grade Lesson Plan
Packet 5/11/2020-5/15/2020
Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
May 11-15, 2020 Course: 10 Art (HS Art II) Teacher(s): Ms. Clare Frank Weekly Plan: Monday, May 11 ⬜ Watch instructional video about a group of murals from the Works Progress Administration. ⬜ Interview an elderly family member; write some notes from the conversation. Tuesday, May 12 ⬜ Watch instructional video about narrative art by Faith Ringgold. ⬜ Interview an elderly family member; write some notes from the conversation. Wednesday, May 13 ⬜ Watch instructional video about Post Office murals. ⬜ Interview an elderly family member; write some notes from the conversation. Thursday, May 14 ⬜ Observe paintings from The Migration Series by Jacob Lawrence. ⬜ Continue your research with your interviews, as needed. Friday, May 15 ⬜ Attend office hours ⬜ Catch-up or review the week’s work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, May 11
1. Watch the instructional video, found as a Material for Monday, May 11. This video features a selection of murals created during the New Deal Era, through the Works Progress Administration.
2. Call an elderly family member, and interview him or her for your “Worst Hard Time” project.*
Tuesday, May 12
1. Watch the instructional video, found as a Material for Tuesday, May 12. This video is about Faith Ringgold’s Story Quilts.
2. Interview an elderly family member for your “Worst Hard Time” project.*
Wednesday, May 13
1. Watch the instructional video, found as a Material for Wednesday, May 13. This video is about American Post Office Murals.
2. Interview an elderly family member for your “Worst Hard Time” project.* Thursday, May 14
1. Follow the link on the Materials tab for Thursday, May 14 to see images from Jacob Lawrence’s Migration Series. Notice the style of his artwork. You may look for just a few minutes or explore the website a bit more, depending on your time and inclination.
2. Continue with your research interviews as needed.
Friday, May 15 Attend office hours or catch up on the week’s work.
* These interviews should be more like conversations. Take notes on your conversation - either during or after, depending on what works best. You may choose to discover stories from more than one elderly relative, or just from one - it is up to you. Consider this daily allotment of time flexible - it is possible you would have two longer conversations rather than four short ones. Also, if you do not have an elderly family member you are able to speak to, you could instead speak with an elderly neighbor or family friend. Please revisit the general overview in the supplemental materials, next page.
“The Worst Hard Time” Storyboard Project: General Overview Your next creative project involves you designing and drawing a storyboard about a phase of family history. For this project you will conduct interviews with elders (oldest members) of your family - people whose memories reach back - and ask them about the worst hard time they remember in their family history, and what they did to get through those times. Research: Whom should you interview? Grandparents or great-grandparents, great-aunts or great-uncles, or elderly friends who are as close to you family as if they are related. How should you interview them? Call them up, chat a bit about you each are doing, and then let them know you want to know about their lives. Listen! Listen, take notes, ask follow up questions. Once you start listening you’ll find people will talk, and some have stories to share that haven’t been shared in a long time, if ever. I used to have an elderly neighbor who’d call me up and ask “Baby, you got a minute?” And if I did, he would talk. I got a picture of rural life century deep South I hadn’t heard before. If you need to bring up the topic of hard times, you might on the first call, or maybe on a follow-up call. You can let them know you are working on a project. Find out what were some of the hardest times they lived through and what they did to get through it. What stories do they have? This week you’ll be conducting interviews and taking notes, and then you’ll start brainstorming. Following that week you will make a storyboard of at least 6 panels. Media and Style: Media will be fairly open on this project - so open a slot in your mind for the ideas to play! There is flexibility on the style, from more realistic to somewhat abstracted, like Jacob Lawrence’s work. However, abstract doesn’t mean “anything goes”. If you look at Lawrence’s work with the principles of design in mind you’ll realize he has great compositions. Definitely your storyboard should not look like a comic strip, which is most of what you will come up with if you google “storyboard”. Even storyboards of family histories are often unimaginative in style and composition. You, on the other hand, are making art! Use the principles of design and elements of art effectively and beautifully, keeping in mind compositional principles you have worked hard to develop! Play to your skills and stretch both your skills and your imagination.
Website for information on Jacob Lawrence’s Migration Series:
https://lawrencemigration.phillipscollection.org/culture/jacob-lawrences-harlem/jacob-lawrence-panel-23-the-migration-spread-from-the-migration-series-1941 This website is posted as a link on a Materials tab for Thursday, May 14.
Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
Week 7: May 11-15, 2020 Course: 10 Chemistry Teacher(s): Ms. Oostindie [email protected] Weekly Plan: Monday, May 11 ⬜ Isotopes review worksheet Tuesday, May 12 ⬜ Read 11.1(pp. 333-334) and take notes ⬜ Balancing nuclear equations worksheet Wednesday, May 13 ⬜ Read 11.2 (pp. 334-335) and take notes ⬜ Watch “History of the Discovery of Radioactivity” video Thursday, May 14 ⬜ Read 11.4 (stopping after gamma emission) and take notes ⬜ Diagram the three types of nuclear decay Friday, May 15 ⬜ Attend office hours ⬜ Catch-up or review the week’s work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, May 11 Complete the attached worksheet titled “Isotopes.” This is a review of material we have already covered in class. If you need to reference your textbook, isotopes can be found in chapter 3, section 3 (pp. 54-55). The key to this worksheet is also attached for you to self-grade your material. *No material will be turned in from this day. Tuesday, May 12 Read 11.1 (pp. 333-334) in your textbook. Your notes should include bolded vocabulary, sample equations, and the differences between chemical and nuclear reactions. Complete the balancing nuclear equations worksheet. You will self-grade the first two questions in a different color pen. *The balancing nuclear equations worksheet will be turned in. Wednesday, May 13 Read 11.2 (pp. 334-335) in your textbook. Your notes should include bolded vocabulary and a drawing of Figure 11.1. In your notes, outline the three major discoveries of the nature of radioactivity. Include the name of the scientist(s), year of discovery, and a summary of their discovery. Watch the “History of the Discovery of Radioactivity” video found on Google Classroom to aid you in your note taking. *No material will be turned in from this day. Thursday, May 14 Read 11.4 through the section titled “Gamma Emission” (pp. 336-339). Your notes should include bolded vocabulary, worked examples, and a nuclear equation for each of the three types of nuclear decay: alpha, beta, and gamma. On a separate sheet of paper, diagram the three types of nuclear decay. You may draw simplified versions of Figure 11.3. Label your diagrams. *Diagrams will be turned in. Friday, May 15 Use this day to attend office hours, catch up on work from this week, scan your documents, and enjoy the start of your weekend! You do not need to include notes in your packet submission, only the documents listed: balancing nuclear equations worksheet and nuclear decay diagrams.
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Balancing Nuclear Equations
Directions: Fill in the missing portion of the following nuclear equations. The first question has been
completed for you as an example.
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Balancing Nuclear Equations
Directions: Fill in the missing portion of the following nuclear equations. The first question has been
completed for you as an example.
Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
Week 7: May 11-15, 2020 Course: Economics, 10th Teacher(s): Mr. Loomis Weekly Plan: Monday, May 11 ⬜ Read part 1 of The Role of Economics, by E. F. Schumacher. Tuesday, May 12 ⬜ Read part 2 of The Role of Economics, by E. F. Schumacher. Wednesday, May 13 ⬜ Write a reflection. Thursday, May 14 ⬜ Continue writing your reflection. Friday, May 15 ⬜ Attend office hours ⬜ Catch-up or review the week’s work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Note: This and next week’s essay are written by E.F. Schumacher, a German-British economist of the early and mid XXth century. In his book Small is Beautiful, from which we are reading essays, he wrestles with many of the ideas that we have been discussing this year: anthropology, value, justice, freedom, meaning, etc. Like Hayek, he is dealing with these issues in the context of the XXth century, which was a very turbulent century: totalitarian regimes, the Great Depression, increase in industrialization, globalization, etc. He offers a critical perspective on the ideas that we have been studying, especially in contrast to Marx and Hayek, and our general study of Microeconomics. He was less extreme than Marx, but less fond of the market than Hayek. His rhetoric is sometimes very strong and it may come off as undermining much of what we have been studying. This is alright. This is part of what it means to think through a subject (in this case Economics) and to ask, as we did on our first day: what is Economics? It is not a very long essay, but it is full of important distinctions. Read carefully and slowly. Pay attention to the reading questions. You do not need to answer all of them, but they are there to help guide you through his argument. Monday, May 11 Read pp. 42 - 48, to the end of the paragraph ending in “...money is the highest of all values.” Reading questions:
- What is Economics and what does an economist do? - What does it mean for Economics to “produce” meaning? Does meaning have a place in the study
of Economics (supply and demand, opportunity cost, marginal utility, etc.)? (p.44b) - Do you think it is a fair representation of Economics to say that:
- the only point of view considered in it is “profit-making,” (p.46t) - and that the market is the “institutionalization of individualism and irresponsibility?”
(p.46b) - What does the author mean by quantitative vs. qualitative? (p.47b)
Tuesday, May 12 Read pp. 48 - 55, starting with “Economics operates legitimately…” Reading questions:
- What does it mean to say that a science is derived from something else? How does this relate to the author’s definition of “meta-economics?” (p.48b)
- What is the significance of the following statement: “Every science is beneficial within its proper limits, but becomes evil and destructive as soon as it transgresses them? (p.49t)
- What is the Greek word for man and what was the name of the first section in this course? Do you think Schumacher would agree with how we started our program of study? (p.49m)
- Is it possible for Economics to be as precise a science as physics? (p.51b)
- Is the author giving a fair account of the fundamental principle of scarcity in Microeconomics? (p.51m, p.54t)
Wednesday, May 13 Today I want you to work on a two-day reflection. You have a total of 40 minutes over two days to write on the question below, labelled Question. Why does the author want us to think about what he calls “meta-economics?” (p.53b, 48b) To be more specific, he claims that we might say that “Economics does not stand on its own two feet, or that it is a derived science–derived from meta-economics.” In distinguishing meta-economics further, he claims that “meta-economics consists of two parts–one dealing with man, and the other dealing with the environment.” He then proceeds to speak about the nature aspect of “meta-economics.” Question: Why does the author want us to focus on a study of nature with respect to the science of Economics? And, why is this distinction essential when it comes to understanding the limitations of the science of Economics? Your response should be about 2 long paragraphs in length, and no more than 1 page. It should be based from within the text; if you need to supplement it you can use our lessons on Microeconomics. However, you do not have very much time, so I do not expect an essay style response. Nevertheless, I do expect you to demonstrate that you have read carefully, thoughtfully, following along with the reading questions. Your response should also contain specific references to the text. Note about handing in the reflection:
- You have two and half options: - You can write it in the Google Doc provide in Google Classroom, or, - You can write it in your own text editor and submit it with the packet, or, - You can write it by hand and scan it in –– this option is not preferable but open if you have
no other option. Thursday, May 14 Continue yesterday’s reflection.
Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.
May 11-14, 2020 Course: 10 Humane Letters Teacher(s): Mr. Garner [email protected] Weekly Plan: Monday, May 11 ⬜ Read Crime and Punishment, Part Five, chapter 5 ⬜ Answer chapter 5 reading questions Tuesday, May 12 ⬜ Read Crime and Punishment, Part Six, chapters 1-2 ⬜ Read through seminar discussion questions Wednesday, May 13 ⬜ Review Tuesday’s reading assignment and seminar questions ⬜ Participate in live seminar Thursday, May 14 ⬜ Read Crime and Punishment, Part Six, chapters 3-4
Monday, May 11
● Read and annotate Part Five chapter 5 carefully. ● Answer the following reading questions in 3-4 complete sentences each.
Crime and Punishment Part Five, chapter 5
1. What do Raskolnikov and Dunya talk about when she comes to see him one last time at his apartment?
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2. Who pays for Katerina Ivanovna’s funeral, and why? _______________________________________________________________________________________________________________________________
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Tuesday, May 12 ● Read and annotate Part Six chapters 1-2 carefully ● Instead of answering reading questions for this reading, spend extra time annotating and thinking
about the reading in preparation for tomorrow’s live seminar discussion. The seminar questions for tomorrow’s discussion are listed below - you do not need to submit written answers to these questions, but I will expect you to come to tomorrow’s seminar prepared to discuss these questions.
Seminar Questions:
1. What is Raskolnikov’s state of mind? To what can we attribute his increased solitude, apathy, and confusion?
2. Why does he feel like a “new man” after his conversation with Razumikhin? 3. Why has Porfiry Petrovich waited so long to arrest Raskolnikov? 4. Porfiry Petrovich seems to have great insight into Raskolnikov’s mind. Are his insights correct?
How does he seem to know Raskolnikov’s mind so well? 5. What does Porfiry Petrovich say about the “need for suffering” in some people? Is Raskolnikov
one of those people who is seeking out suffering for himself? a. Connected to this discussion of suffering, Porfiry, like Sonya, tells Raskolnikov that he
should embrace suffering. Does Porfiry mean the same thing by this as Sonya? 6. Porfiry Petrovich suggests that what Raskolnikov needs is “a change of air.” (460) Recall that
Svidrigailov suggested the same thing to Raskolnikov in the last chapter. What do they mean by this?
7. Is Raskolnikov penitent yet for his crime?
Wednesday, May 13 ● Review yesterday’s reading and annotations, as well as the list of seminar questions. ● Participate in the live seminar! The link for the Zoom meeting can be found on the Google
Classroom page.
Thursday, May 14 ● Read and annotate Part Six chapters 3-4 carefully, paying special attention to the following points:
○ In these chapters the character of Svidrigailov comes to the forefront again. Continue to consider his earlier claim, that he and Raskolnikov were alike at heart; do they seem more alike now as they converse? Why is Raskolnikov so repulsed by the possibility of their alikeness?
○ Svidrigailov talks about “depravity” in his conversation with Raskolnikov. Consider whether Svidrigailov’s depravity is different than Raskolnikov’s, and if one form of depravity is worse than the other.
○ Svidrigailov begins to appear in this chapter as a “Napoleon” of sorts - that is to say, it seems as though he, much more than Raskolnikov, has abandoned the conventional constraints of morality and is quite comfortable “stepping over” whenever he feels like it. Is this the sort of man Raskolnikov thought he might become?
Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
Week 7: May 11-15, 2020 Course: 10 Latin IV Teacher(s): Ms. Mueller [email protected] Supplemental Links: Aeneid I.102-123 Online Grammar Reference
Aeneid Online Vocabulary Reference Weekly Plan:
Monday, May 11 ⬜ Check answer keys to last week’s Aeneid I. 113-123 worksheet and translation ⬜ Prepare for Tuesday’s and Wendesday’s Aeneid I.34-123 Assessment ⬜ Log into Google Classroom and watch an instructional video on lines I.102-123
Tuesday, May 12 ⬜ Log into Google Classroom and complete “Aeneid I. 34-123 Assessment: Part I”
Wednesday, May 13 ⬜ Log into Google Classroom and complete “Aeneid I. 34-123 Assessment: Part I”
Thursday, May 14 ⬜ Read the attached translation of Aeneid I. 124-156 ⬜ Read Aeneid lines I.57-58, and 170-179 in Latin (pp. 26-28) and the translation of lines 159-169 ⬜ Complete “Aeneid I. 157-158 and 170-179 Questions” worksheet
Friday, May 15 ⬜ No new assignments, attend office hours and/or get caught up on previous work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, May 11 1. Review the answer keys for last week’s “Aeneid I. 113-123 Grammar and Reading Questions”
and the Aeneid I. 113-123 translation for the Aeneid I.34-123 Assessment tomorrow and Wednesday. If you still have access to those assignments from last week, I encourage you to compare your answers to those on the answer keys.
The assessment tomorrow and Wednesday will be open book and open note, but as you review today, you want to make sure you are familiar enough with lines I.34-123 to:
a. answer reading comprehension and grammar questions in a multiple choice or true/ false format
b. identify the use of literary and rhetorical devices c. scan lines of dactylic hexameter d. write a thoughtful essay supported by the Latin text in response to a specific
prompt
2. Log into Google Classroom and watch an instructional video guiding you through lines I.102-123 of the Aeneid. Tuesday, May 12
1. Take some time to get out the materials you will want to reference for the open book/open note “Aeneid I.34-123 Assessment: Part I.” This part of the assessment will test reading comprehension, grammar analysis, recognition of literary devices, and scansion. Please note that while you can reference any of your notes or materials, you may not ask for or receive help from anyone during the assessment.
2. Log into Google Classroom and complete the “Aeneid I. 34-123 Assessment: Part I.”
Wednesday, May 13 1. Take some time to get out the materials you will want to reference for the open book/open note
“Aeneid I.34-123 Assessment: Part II.” For this part of the assessment, you will write a short essay responding to a prompt over a specific passage from lines 34-123. Your essay should be well developed and make frequent references to the Latin passage demonstrating your ability to understand the Latin as well as offer thoughtful analysis on the passage as a piece of literature.
2. Log into Google Classroom and complete the “Aeneid I. 34-123 Assessment: Part II.”
Thursday, May 14 1. Read the attached translation of Book I. 124-156 2. Read lines I. 157-158 and 170-179 in Latin and a translation of lines I. 159-169. Complete the
“Aeneid I. 157-158 and 170-179 Questions” worksheet.
Friday, May 15 No new assignments! Use this day to attend office hours and/or get caught up on previous work from the week!
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__________________________ Aeneid I.113-123 Reading and Grammar Questions __________________________
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I. Comprehension Questions: Answer the following questions about lines 113-123.
1. What happens to the ship that is carrying Orontes (lines 113-115)?
A huge wave strikes against the ship.
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2. What happens to the helmsman of Orontes’ ship (lines 115-117)?
The helmsman is cast out of the ship and is turned onto his head.
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3. Name three things that appear among the waves in lines 118-119.
1. Scattered men (swimming) 2. the men’s weapons 3. Trojan wealth
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4. Vergil mentions four more ships that were damaged in the storm. With what comrade was each associated (lines 120-123)?
Ilioneus, brave Achates, Abas, and old Aletes.
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II. Grammar Questions: Indicate True or False by marking a “T” or an “F” in the space provided.
1. F ipsius (line 114) refers to Oronten (line 113). ipsius refers to Aeneas
2. T Line 118 can be translated “There appear scattered men floating in the huge whirlpool.”
3. F The object of vicit in line 122 is hiems (line 122). hiems is the subject of vicit
4. F In line 122, laterum is accusative. laterum is genitive plural
III. Scansion: After watching the instructional video on Google Classroom, scan the following lines of dactylic hexameter.
Iam vali│d(am) Īlio │neī nā │vem, iam │fortis A│chātae, et quā │vectus A│bās, et │quā gran │daevus A│lētēs, vīcit hi │ems; la│xīs late │rum com│pāgibus │ omnēs accipi│unt ini│mīc (um) imb│rem rī │mīsque fa │tīscunt.
Aeneid Book I.113-123 Translation (Lines 113-117) One (ship), which was carrying the Lycians and faithful Orontes, before the eyes of (Aeneas) himself, a huge wave from its summit strikes against the stern: the helmsman is cast out and leaning forward is turned onto his head, but a wave twists that (ship) three times in the same place driving (it) around and a swift whirlpool swallows (it) in the sea. (Lines 118-119) There appear through the waves scattered men floating in the vast whirlpool, the weapons of men and planks and Trojan wealth. (Lines 120-123) Now the mighty ship of Ilioneus, now (the ship) of brave Achates, and (the ship) in which Abas is carried, and (the ship) in which old Aletes (is carried) the storm conquered; all (the ships) receive the hostile flood water with the loose seams of their sides and they gape with fissures.
Translation of Aeneid Book I.124-156 (by A. S. Kline)
BkI:124-156 Neptune Intervenes
Neptune, meanwhile, greatly troubled, saw that the sea
was churned with vast murmur, and the storm was loose
and the still waters welled from their deepest levels:
he raised his calm face from the waves, gazing over the deep.
He sees Aeneas’s fleet scattered all over the ocean,
the Trojans crushed by the breakers, and the plummeting sky.
And Juno’s anger, and her stratagems, do not escape her brother.
He calls the East and West winds to him, and then says:
‘Does confidence in your birth fill you so? Winds, do you dare,
without my intent, to mix earth with sky, and cause such trouble,
now? You whom I – ! But it’s better to calm the running waves:
you’ll answer to me later for this misfortune, with a different punishment.
Hurry, fly now, and say this to your king:
control of the ocean, and the fierce trident, were given to me,
by lot, and not to him. He owns the wild rocks, home to you,
and yours, East Wind: let Aeolus officiate in his palace,
and be king in the closed prison of the winds.’
So he speaks, and swifter than his speech, he calms the swollen sea,
scatters the gathered cloud, and brings back the sun.
Cymothoë and Triton, working together, thrust the ships
from the sharp reef: Neptune himself raises them with his trident,
parts the vast quicksand, tempers the flood,
and glides on weightless wheels, over the tops of the waves.
As often, when rebellion breaks out in a great nation,
and the common rabble rage with passion, and soon stones
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and fiery torches fly (frenzy supplying weapons),
if they then see a man of great virtue, and weighty service,
they are silent, and stand there listening attentively:
he sways their passions with his words and soothes their hearts:
so all the uproar of the ocean died, as soon as their father,
gazing over the water, carried through the clear sky, wheeled
his horses, and gave them their head, flying behind in his chariot.
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Translation of Aeneid Book I.159-169 (by A. S. Kline)
BkI:159-169 Shelter on the Libyan Coast
There is a place there in a deep inlet: an island forms a harbour
with the barrier of its bulk, on which every wave from the deep
breaks, and divides into diminishing ripples.
On this side and that, vast cliffs and twin crags loom in the sky,
under whose summits the whole sea is calm, far and wide:
then, above that, is a scene of glittering woods,
and a dark grove overhangs the water, with leafy shade:
under the headland opposite is a cave, curtained with rock,
inside it, fresh water, and seats of natural stone,
the home of Nymphs. No hawsers moor the weary ships
here, no anchor, with its hooked flukes, fastens them.
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__________________________ Aeneid I.157-158 and 170-179 Questions __________________________
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I. Comprehension Questions: Answer the following questions about lines 157-158 and 170-179.
1. Why do Aeneas and his followers end up on the shores of Libya (lines 157-158)?
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2. What does the phrase magno telluris amore (line 171) tell us about the shipwrecked Trojans?
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3. What is Achates doing in lines 174-176?
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4. Why do you think Vergil goes into such detail describing the preparation of the food in lines 174-179?
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II. Answer the following multiple choice questions on lines 157-158 and 170-179 .
1. The best translation of lines 157-158 (Defessi . . . oras) is
a. Aeneas’s tired followers strive toward the shores in their course, which is very near, and they are turned toward Libya’s coast
b. Aeneas’s tired followers, who are nearest to the shore in their course, aim toward it, and they are turned toward Libya’s coast.
c. The weary followers of Aeneas strive to seek with their course the shores which are nearest, and they are turned toward the coast of Libya.
d. The weary followers of Aeneas seek in their haste the nearest shores, which they strive toward, and they are turned toward the coast of Libya.
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2. In line 70, omni modifies
a. huc (line 70)
b. septem (line 170)
c. numero (line 171)
d. amore (line 171)
3. A figure of speech that occurs in line 177 is
a. personification
b. anaphora
c. litotes
d. metonymy
4. The metrical pattern of the first four feet of line 179 is
a. dactyl-spondee-dactyl-spondee
b. spondee-dactyl-spondee-dactyl
c. spondee-dactyl-spondee-spondee
d. dactyl-dactyl-spondee-dactyl
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Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
Week 7: May 11-15, 2020 Course: 10 Precalculus Teacher(s): Mr. Simmons Weekly Plan: Monday, May 11 ⬜ Story time! ⬜ Check out the “Trig Cheat Sheet.” ⬜ Check answers from previous assignments. Tuesday, May 12 ⬜ Complete problems 1-8 (A-G) from “Polar Plane.” Wednesday, May 13 ⬜ Check answers. (An answer key will be posted.) Thursday, May 14 ⬜ Read “The Unit Circle.” Friday, May 15 ⬜ Attend office hours ⬜ Catch up or review the week’s work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. ___________________________________ Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _____________________________________ Parent Signature
Monday, May 11 Happy Monday! I hope everyone’s doing well.
1. Story time! If technologically feasible, please email me and let me know how you’re doing. I love hearing from each and every one of you.
I heard from a few students (thanks so much for filling out the survey!), and one comment about the readings was that the information from each section wasn’t all available in one place for reference while working problems. I hope the “Trig Cheat Sheet” included in this packet will help. It has all the formulas you’re learning, and then some. (Don’t feel like you need to go memorize the ones that the text I gave you doesn’t cover.) I am also including answer keys for all previous problem sets (sorry I didn’t have those earlier). From now on I’ll post an answer key the day after each problem set. So:
2. At least glance at the cheat sheet. Better yet, print it out. Have it at the ready. Maybe frame it. Put it on your wall.
3. Check your answers to all previous problems using the answer keys included herein. Tuesday, May 12
1. Complete Problems 1-8 from “Introduction to the Polar Plane” (pp. 144-46). For some of the problems, there are multiple exercises, labeled with capital letters. For each of those types of problems, please complete only A through G. (That means you’re still doing every exercise labeled in small roman numerals.)
Wednesday, May 13
1. Check your answers from yesterday. (I’ll post the answer key.) Thursday, May 14
1. Read “The Unit Circle.” Remember to read slowly! Try to work each of the example problems yourself before looking at the right answer.
© 2005 Paul Dawkins
Trig Cheat Sheet
Definition of the Trig Functions Right triangle definition For this definition we assume that
02π
θ< < or 0 90θ° < < ° .
oppositesin
hypotenuseθ = hypotenusecsc
oppositeθ =
adjacentcoshypotenuse
θ = hypotenusesecadjacent
θ =
oppositetanadjacent
θ = adjacentcotopposite
θ =
Unit circle definition For this definition θ is any angle.
sin
1y yθ = = 1csc
yθ =
cos1x xθ = = 1sec
xθ =
tan yx
θ = cot xy
θ =
Facts and Properties Domain The domain is all the values of θ that can be plugged into the function. sinθ , θ can be any angle cosθ , θ can be any angle
tanθ , 1 , 0, 1, 2,2
n nθ π ≠ + = ± ±
…
cscθ , , 0, 1, 2,n nθ π≠ = ± ± …
secθ , 1 , 0, 1, 2,2
n nθ π ≠ + = ± ±
…
cotθ , , 0, 1, 2,n nθ π≠ = ± ± … Range The range is all possible values to get out of the function.
1 sin 1θ− ≤ ≤ csc 1 and csc 1θ θ≥ ≤ − 1 cos 1θ− ≤ ≤ sec 1 andsec 1θ θ≥ ≤ −
tanθ−∞ < < ∞ cotθ−∞ < < ∞
Period The period of a function is the number, T, such that ( ) ( )f T fθ θ+ = . So, if ω is a fixed number and θ is any angle we have the following periods.
( )sin ωθ → 2T πω
=
( )cos ωθ → 2T πω
=
( )tan ωθ → T πω
=
( )csc ωθ → 2T πω
=
( )sec ωθ → 2T πω
=
( )cot ωθ → T πω
=
θ adjacent
opposite hypotenuse
x
y
( ),x y
θ
x
y 1
© 2005 Paul Dawkins
Formulas and Identities Tangent and Cotangent Identities
sin costan cotcos sin
θ θθ θ
θ θ= =
Reciprocal Identities 1 1csc sin
sin csc1 1sec cos
cos sec1 1cot tan
tan cot
θ θθ θ
θ θθ θ
θ θθ θ
= =
= =
= =
Pythagorean Identities 2 2
2 2
2 2
sin cos 1tan 1 sec1 cot csc
θ θ
θ θ
θ θ
+ =
+ =
+ =
Even/Odd Formulas ( ) ( )( ) ( )( ) ( )
sin sin csc csc
cos cos sec sec
tan tan cot cot
θ θ θ θ
θ θ θ θ
θ θ θ θ
− = − − = −
− = − =
− = − − = −
Periodic Formulas If n is an integer.
( ) ( )( ) ( )( ) ( )
sin 2 sin csc 2 csc
cos 2 cos sec 2 sec
tan tan cot cot
n n
n n
n n
θ π θ θ π θ
θ π θ θ π θ
θ π θ θ π θ
+ = + =
+ = + =
+ = + =Double Angle Formulas
( )( )
( )
2 2
2
2
2
sin 2 2sin cos
cos 2 cos sin
2cos 11 2sin
2 tantan 21 tan
θ θ θ
θ θ θ
θ
θθ
θθ
=
= −
= −
= −
=−
Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then
180and 180 180
t x tt xx
π ππ
= ⇒ = =
Half Angle Formulas (alternate form)
( )( )
( )( )( )( )
2
2
2
1 cos 1sin sin 1 cos 22 2 2
1 cos 1cos cos 1 cos 22 2 2
1 cos 21 costan tan2 1 cos 1 cos 2
θ θθ θ
θ θθ θ
θθ θθ
θ θ
−= ± = −
+= ± = +
−−= ± =
+ +Sum and Difference Formulas
( )( )
( )
sin sin cos cos sin
cos cos cos sin sintan tantan
1 tan tan
α β α β α β
α β α β α β
α βα β
α β
± = ±
± =
±± =
∓
∓
Product to Sum Formulas
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1sin sin cos cos21cos cos cos cos21sin cos sin sin21cos sin sin sin2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
= − − +
= − + +
= + + −
= + − −
Sum to Product Formulas
sin sin 2sin cos2 2
sin sin 2cos sin2 2
cos cos 2cos cos2 2
cos cos 2sin sin2 2
α β α βα β
α β α βα β
α β α βα β
α β α βα β
+ − + =
+ − − =
+ − + =
+ − − = −
Cofunction Formulas
sin cos cos sin2 2
csc sec sec csc2 2
tan cot cot tan2 2
π πθ θ θ θ
π πθ θ θ θ
π πθ θ θ θ
− = − = − = − = − = − =
© 2005 Paul Dawkins
Unit Circle
For any ordered pair on the unit circle ( ),x y : cos xθ = and sin yθ = Example
5 1 5 3cos sin3 2 3 2π π = = −
3π
4π
6π
2 2,2 2
3 1,2 2
1 3,2 2
60°
45°
30°
23π
34π
56π
76π
54π
43π
116π
74π
53π
2π
π
32π
0 2π
1 3,2 2
−
2 2,2 2
−
3 1,2 2
−
3 1,2 2
− −
2 2,2 2
− −
1 3,2 2
− −
3 1,2 2
−
2 2,2 2
−
1 3,2 2
−
( )0,1
( )0, 1−
( )1,0−
90° 120°
135°
150°
180°
210°
225°
240° 270°
300° 315°
330°
360°
0° x
( )1,0
y
© 2005 Paul Dawkins
Inverse Trig Functions Definition
1
1
1
sin is equivalent to sincos is equivalent to costan is equivalent to tan
y x x yy x x yy x x y
−
−
−
= =
= =
= =
Domain and Range
Function Domain Range 1siny x−= 1 1x− ≤ ≤
2 2yπ π
− ≤ ≤
1cosy x−= 1 1x− ≤ ≤ 0 y π≤ ≤
1tany x−= x−∞ < < ∞ 2 2
yπ π− < <
Inverse Properties ( )( ) ( )( )
( )( ) ( )( )( )( ) ( )( )
1 1
1 1
1 1
cos cos cos cos
sin sin sin sin
tan tan tan tan
x x
x x
x x
θ θ
θ θ
θ θ
− −
− −
− −
= =
= =
= =
Alternate Notation
1
1
1
sin arcsincos arccostan arctan
x xx xx x
−
−
−
=
=
=
Law of Sines, Cosines and Tangents
Law of Sines sin sin sin
a b cα β γ
= =
Law of Cosines 2 2 2
2 2 2
2 2 2
2 cos2 cos2 cos
a b c bcb a c acc a b ab
α
β
γ
= + −
= + −
= + −
Mollweide’s Formula ( )1
212
cossin
a bc
α βγ−+
=
Law of Tangents ( )( )( )( )( )( )
1212
1212
1212
tantan
tantan
tantan
a ba b
b cb c
a ca c
α βα β
β γβ γ
α γα γ
−−=
+ +
−−=
+ +
−−=
+ +
c a
b
α
β
γ
4.1.1�5 odd � Answer Key
Precalculus
Mr. Simmons
1. (a)
122 + 152 = c2
144 + 225 = c2
369 = c2
c =√369
= 3√41
≈ 19.21.
(b) c =√6.54 ≈ 2.56
(c) c =√640 = 8
√10 ≈ 25.30
(d) b =√204.75 ≈ 14.31
(e) a =√27632 = 4
√1727 ≈ 166.23
(f) b =√19.25 ≈ 4.39
(g) a = 3√17 ≈ 12.37
(h) c =√101 ≈ 10.05
3. No, not all triangles are possible. You can-not have a right triangle with legs of length10 and 20 and a hypotenuse of 15, because102 + 202 6= 152. Even outside of right tri-angles, not every triangle is possible. Trydrawing a triangle with side lengths 5, 10,and 20.
5. (a) c = 5
(b) a = 5
(c) a = 7, b = 24, c = 25; a = 8, b =15, c = 17; a = 9, b = 40, c = 41;a = 11, b = 60, c = 61; a = 12, b =35, c = 37; a = 13, b = 84, c = 85(and there are others).
1
4.2.1�5 � Answer Key
Precalculus
Mr. Simmons
1. (a) sinα = 3/5,cosα = 4/5,tanα = 3/4;sinβ = 4/5,cosβ = 3/5,tanβ = 4/3.
(b) sinα = 1/√82 ≈ 0.11,
cosα = 9/√82 ≈ 0.99,
tanα = 1/9;sinβ = 9/
√82 ≈ 0.99,
cosβ = 1/√82 ≈ 0.11,
tanβ = 9.
(c) sinα = 2/√5 ≈ 0.89,
cosα = 1/√5 ≈ 0.45,
tanα = 2;sinβ = 1/
√5 ≈ 0.45,
cosβ = 2/√5 ≈ 0.89,
tanβ = 1/2.
(d) sinα = 2/4.5 ≈ 0.44,cosα = 4/4.5 ≈ 0.89,tanα = 1/2;sinβ = 4/4.5 ≈ 0.89,cosβ = 2/4.5 ≈ 0.44,tanβ = 2.
2. (a) The �really nice angles� are 30◦, 45◦,and 60◦.
(b) [Drawing of any two right trianglewith acute angles 30◦ and 60◦ or both45◦. Such triangles could have a vari-ety of side lengths, two examples be-ing side lenths 1-1-
√2 and 3-4-5.]
(c)
α sinα cosα tanα
30◦ 12
√32
√33
45◦√22
√22 1
60◦√32
12
√3
3. (a) It will decrease.
(b) Zero.
(c) Zero.
(d) cosα = 7/7.04 ≈ 0.99
(e) The length of the hypotenuse will ap-proach the length of the adjacent side.
(f) One.
4. (a) An example of such a triangle isone with side lengths 1-10-
√101.
(√101 ≈ 10.05.) Here, α will be 84◦.
(b) sin 90◦ = 1; cos 90◦ = 0.
5. Adding to the earlier table and approxi-mating when appropriate, we have the fol-lowing table.
α sinα cosα tanα
0◦ 0 1 05◦ ∼ 0.09 ∼ 0.996 ∼ 0.0910◦ ∼ 0.17 ∼ 0.98 ∼ 0.1815◦ ∼ 0.25 ∼ 0.97 ∼ 0.2720◦ ∼ 0.34 ∼ 0.94 ∼ 0.3625◦ ∼ 0.42 ∼ 0.91 ∼ 0.47
30◦ 12 = 0.5
√32 ≈ 0.87
√33 ≈ 0.58
35◦ ∼ 0.57 ∼ 0.82 ∼ 0.7040◦ ∼ 0.64 ∼ 0.77 ∼ 0.84
45◦√22 ≈ 0.71
√22 ≈ 0.71 1
50◦ ∼ 0.77 ∼ 0.64 ∼ 1.1955◦ ∼ 0.82 ∼ 0.57 ∼ 1.43
60◦√32 ≈ 0.87 1
2 = 0.5√3 ≈ 1.73
65◦ ∼ 0.91 ∼ 0.42 ∼ 2.1470◦ ∼ 0.94 ∼ 0.34 ∼ 2.7575◦ ∼ 0.97 ∼ 0.25 ∼ 3.7380◦ ∼ 0.98 ∼ 0.17 ∼ 5.6785◦ ∼ 0.996 ∼ 0.09 ∼ 11.4390◦ 1 0 unde�ned
1
4.3.1�14,16 � Answer Key
Precalculus
Mr. Simmons
1. (a) sinα = 12/13,cosα = 5/13,tanα = 12/5,cscα = 13/12,secα = 13/5,cotα = 5/12.
(b) sinα = 4/5,cosα = 3/5,tanα = 4/3,cscα = 5/4,secα = 5/3,cotα = 3/4.
(c) sinα = 24/25,cosα = 7/25,tanα = 24/7,cscα = 25/24,secα = 25/7,cotα = 7/24.
(d) sinα = 1/4,cosα =
√17/4,
tanα = 1/√17,
cscα = 4,secα = 4/
√17,
cotα =√17.
(e) sinα =√69/13,
cosα = 10/13,tanα =
√69/10,
cscα = 13/√69,
secα = 13/10,cotα = 10/
√69.
(f) sinα = 5/√61,
cosα = 6/√61,
tanα = 5/6,cscα =
√61/5,
secα =√61/5,
cotα = 6/5.
(g) sinα = 2/7,cosα = 3
√5/7,
tanα = 2/(3√5),
cscα = 7/2,secα = 7/
(3√5),
cotα = 3√5/2.
(h) sinα =√51/10,
cosα = 7/10,tanα =
√51/7,
cscα = 10/√51,
secα = 10/7,cotα = 7/
√51.
2. The triangle in Example 1c has oppositeleg length a and hypotenuse length 1. Ifwe call the adjacent leg length b, we havea2 + b2 = 12, so b =
√1− a2. So the six
trig ratios are as follows:sinα = a,cosα =
√1− a2,
tanα = a/√1− a2,
cscα = 1/a,secα = 1/
√1− a2,
cotα =√1− a2/a.
3. (a) If tanα = d, then we can consider thetriangle with leg lengths d and 1, withhypotenuse
√d2 + 1. Then have the
following trig ratios:sinα = d/
√d2 + 1,
cosα = 1/√d2 + 1,
tanα = d,cscα =
√d2 + 1/d,
secα =√d2 + 1,
cotα = 1/d.
4. (a) sinα = a/c,cosα = b/c,
1
2
tanα = a/b,cscα = c/asecα = c/bcotα = b/a.
5. (a) sin 30◦ = cos 60◦
(b) cos 10◦ = sin 80◦
(c) cot 7◦ = tan 83◦
(d) sec 64◦ = csc 26◦
(e) cos 31◦ = sin 59◦
(f) tan 14◦ = cot 76◦
(g) csc 47◦ = sec 43◦
(h) sin 25◦ = cos 65◦
(i) tan (β + γ) = cot (90◦ − β − γ)(j) sinβ = cos (90◦ − β)
6. sin (90◦ − θ) = cos θ,cos (90◦ − θ) = sin θ,csc (90◦ − θ) = sec θ,sec (90◦ − θ) = csc θ,tan (90◦ − θ) = cot θ,cot (90◦ − θ) = tan θ.
7. csc θ = 1/ sin θ,sin θ = 1/ csc θ,sec θ = 1/ cos θ,cos θ = 1/ sec θ,cot θ = 1/ tan θ,tan θ = 1/ cot θ.
8. (a) sin2 30◦ = (1/2)2= 1/4
(b) cos2 30◦ =(√
3/2)2
= 3/4
(c) tan2 30◦ =(√
3/3)2
= 1/3
(d) cos2 45◦ =(√
2/2)2
= 1/2
(e) tan2 45◦ = 12 = 1
(f) sin2 60◦ =(√
3/2)2
= 3/4
(g) cos2 60◦ = (1/2)2= 1/4
(h) tan2 60◦ =√32= 3
(i) sin 30◦2 = sin 900◦ = unde�ned (un-til we have the unit-circle de�nition ofsine)
(j) cos2 α = (cosα)2
9.
α sin2 α cos2 α tan2 α
0◦ 0 1 05◦ ∼ 0.01 ∼ 0.99 ∼ 0.0110◦ ∼ 0.03 ∼ 0.93 ∼ 0.0315◦ ∼ 0.07 ∼ 0.88 ∼ 0.0720◦ ∼ 0.12 ∼ 0.82 ∼ 0.1325◦ ∼ 0.18 ∼ 0.67 ∼ 0.2230◦ 1
4 = 0.25 34 = 0.75 1
3 ≈ 0.3335◦ ∼ 0.33 ∼ 0.59 ∼ 0.4940◦ ∼ 0.41 ∼ 0.59 ∼ 0.7045◦ 1
2 = 0.5 12 = 0.5 1
50◦ ∼ 0.59 ∼ 0.41 ∼ 1.4255◦ ∼ 0.67 ∼ 0.33 ∼ 2.0460◦ 3
4 = 0.75 14 = 0.25 3
65◦ ∼ 0.82 ∼ 0.18 ∼ 4.6070◦ ∼ 0.88 ∼ 0.12 ∼ 7.5575◦ ∼ 0.93 ∼ 0.07 ∼ 13.9380◦ ∼ 0.97 ∼ 0.03 ∼ 32.1685◦ ∼ 0.99 ∼ 0.01 ∼ 130.6590◦ 1 0 unde�ned
10. (a) 1; 1; no maximum
(b) 0; 0; 0
(c) The sin and sin2 functions have thesame shape, but sin2 has only pos-itive values and is shorter. It alsohas hills/valleys twice as frequently.(We say that its �period� is half thelength.)
11. sin2 θ + cos2 θ = 1
12. These expressions are all of the formsin2 θ + cos2 θ, so they all equal one.
13. Since sin2 θ + cos2 θ = 1, we have
sin2 θ = 1− cos2 θ
and
cos2 θ = 1− sin2 θ.
3
14. (a)
tanα · cscα =sinα
cosα· 1
sinα
=1
cosα= secα.
(b)
(sinα+ cosα)2= sin2 α+ 2 sinα cosα+ cos2 α
= 2 sinα cosα+ 1.
16. (a) True. Although this is not the cofunc-tion identity (that would be sinα =cos (90◦ − α)), the statement here is
still true because
sinα = cos (90◦ − α)= cos (− [90◦ − α])= cos (α− 90◦) .
This fact is, however, dependent onthe unit-circle de�nition of cosine, so,strictly speaking, since cosine is un-de�ned for negative angle measuresin right-triangle trigonometry, �false�would also be an acceptable answer,as long as you understood the reason.
(b) False. Counterexample: α = β =30◦.
(c) False. Counterexample: α = 5◦
(d) False. Squares are never negative.
5.1.1�14 � Answer Key
Precalculus
Mr. Simmons
1. (a) 2
(b) 1/3 ≈ 0.33
(c) 20/3 ≈ 6.67
(d) 64/3 ≈ 21.33
2. (a) 10
(b) 9
(c) 5/4 = 1.25
(d) 5
3. (a) 5
(b) 1
(c) 10/3 ≈ 3.33
(d) 12
4. See 1, 2, and 3 sketched on pp. 129�30.Come to o�ce hours if you want to see therest sketched.
5. (a) 10π/9
(b) 5π/9
(c) 5π/18
(d) 270/π◦
(e) 15◦
(f) 4π
(g) 36◦
(h) 180◦
6. (a) π/6, π/4, π/3, π/4,π/2,3π/2,2π.
(b) π/2
(c) 120◦, 135◦, 150◦, 210◦, 225◦, 240◦,300◦, 315◦, 330◦.
(d) 2π/3, 3π/4, 5π/6, 7π/6, 5π/4, 4π/3,5π/3, 7π/4, 11π/6.
7. (a) 1/2
(b) The more pieces you divide a wholeinto, by necessity the smaller thepieces will be (if they are of equalsize�if they are unequally sized, stillthe average size will be smaller).
(c) π/3
(d) 2
(e) π
(f) 15π/4
8. A half rotation would be π, which, writtenas 6π/6, is obviously greater than 5π/6.
9. Quadrant IV. One way to tell quickly isthat 11π/6 is very nearly 2π (which fact ismade obvious by writing 2π as 12π/6).
10. 5π/3 = 300◦ < 315◦ = 7π/4
11.5π
3=
20π
12<
21π
12=
7π
4
12. (a) 24π in
(b) 12π in
(c) 120π in
(d) 3 (radians)
13. (a) 10 in
(b) 10π (radians)
14. (a) 60/ (13π) ft ≈ 1.47 ft = 17.64 in
(b) 80/3 ft ≈ 26.67 ft = 320.04 in
1
Uni
t fiv
e –
144
§𝟐 E
xerc
ises
1.)
Plo
t the
follo
win
g po
ints
. You
may
use
the
sam
e Po
lar P
lane
if yo
u w
ish.
(A) 𝐴( 2
,30
°)
(B) 𝐵( 1
,45
°)
(C) 𝐶( 3
,30
0°)
(D
) 𝐷( 2
,15
0°)
(E
) 𝐸( 4
,31
5°)
(F
) 𝐹( 2
,21
0°)
(G) 𝐺( 0
.5,6
0°)
(H
) 𝐻( 3
.5,1
35
°)
(I) 𝐼(2
0,1
80
°)
(J) 𝐽(
1,2
60
°)
(K) 𝐾( 3
.5,0
°)
(L)
𝐿(7 2
,35
0°)
2.
) Be
fore
you
sta
rt p
lott
ing
poin
ts u
sing
radi
ans,
crea
te a
Pol
ar P
lane
just
like
we
did
in t
his
unit,
exc
ept
inst
ead
of u
sing
30
°,4
5°,
… u
se t
he c
orre
spon
ding
rad
ian
mea
sure
s. M
ake
sure
eac
h an
gle
is la
bele
d.
3.)
Now
that
you
hav
e a
Pola
r Pla
ne w
ith ra
dian
s, pl
ot th
e fo
llow
ing
poin
ts.
(A) 𝐴(2
,𝜋 2)
(B) 𝐵(3
,𝜋 3)
(C) 𝐶(4
,2𝜋 3)
(D) 𝐷(1
,5𝜋 4)
(E) 𝐸( 4
,1)
(F) 𝐹(3
,11𝜋
6)
(G) 𝐺( 2
,𝜋)
(H) 𝐻(2
,3𝜋 4)
(I) 𝐼(2
,𝜋 6)
(J) 𝐽(3
00
,3𝜋 2)
(K) 𝐾(5
,5𝜋 3)
(L)
𝐿(2
,7𝜋 4)
(M) 𝑀(3
,7𝜋 6)
4.)
Now
let’s
wor
k w
ith n
egat
ive
angl
es.
(A)
Wha
t doe
s a
nega
tive
angl
e re
pres
ent,
or te
ll yo
u to
do?
(B
) N
ow c
reat
e a
Pola
r Pla
ne ju
st li
ke y
ou d
id in
2.),
exc
ept t
his
time
labe
l eac
h an
gle
as −
30
°,−4
5°,
… a
s w
e st
arte
d to
do
in th
e re
adin
g. M
ake
sure
eac
h an
gle
is la
bele
d.
(C)
Crea
te a
noth
er P
olar
Pla
ne, e
xcep
t thi
s tim
e us
e ne
gativ
e ra
dian
mea
sure
s. Ag
ain,
mak
e su
re e
ach
angl
e is
labe
led.
5.
) Pl
ot th
e fo
llow
ing
poin
ts.
(A) 𝐴( 2
,−4
5°)
(B
) 𝐵(3
,−2𝜋 3)
(C) 𝐶( 1
,−1
80
°)
(D) 𝐷(2
,−7𝜋 4)
(E) 𝐸( 3
,−3
00
°)
(F) 𝐹(4
,−5𝜋 6)
(G) 𝐺( 5
,−2
5°)
(H
) 𝐻(2
,−2)
(I) 𝐼(
3,−
10
0°)
(J)
𝐽(4
,−2𝜋)
(K) 𝐾( 3
,−1
35
°)
(L)
𝐿(2
,−3𝜋 4)
6.)
Now
let’s
wor
k w
ith c
oter
min
al a
ngle
s. Th
is is
incr
edib
ly im
port
ant f
or o
ur w
ork
in
trig
onom
etry
, and
was
one
of t
he m
ain
reas
ons
we
chos
e to
wor
k w
ith th
e Po
lar
Plan
e be
fore
wor
king
with
the
Uni
t Circ
le. U
se th
e Po
lar
Plan
es y
ou c
reat
ed fr
om
prev
ious
exe
rcise
s to
hel
p yo
u.
§2
Intr
oduc
tion
to th
e Po
lar P
lane
– 1
45
(A)
Giv
e tw
o di
ffere
nt c
oter
min
al a
ngle
s (in
deg
rees
) of
the
giv
en a
ngle
m
easu
re.
i. 3
0°.
ii. 4
5°
iii.
90
° iv
. 1
35
° v.
1
50
° vi
. 1
80
°
vii. 2
00
° vi
ii.
10
° ix
. 0
° x.
3
30
° xi
. 24
0°
xii.
12
0°
(B)
Giv
e tw
o di
ffere
nt c
oter
min
al a
ngle
s (in
radi
ans)
of t
he g
iven
ang
le m
easu
re.
i. 𝜋 6
ii. 𝜋 4
iii.
𝜋 3
iv.
𝜋 2
v.
2𝜋 3
vi.
3𝜋 4
vii.
5𝜋 6
viii.
𝜋
ix.
7𝜋 6
x. 5𝜋 4
xi.
4𝜋 3
xii.
3𝜋 2
xiii.
5𝜋 3
xiv.
7𝜋 4
xv.
11𝜋
6
xvi. 2𝜋
7.
) Le
t us
now
exp
lore
how
to c
onve
rt a
Pol
ar p
oint
into
a n
orm
al re
ctan
gula
r poi
nt.
Cons
ider
the
Fig
ure
belo
w, o
f 𝐴( 3
,30
°). W
e w
ill c
reat
e a
tria
ngle
usin
g th
is po
int
and
the
Pole
.
Not
ice
that
if w
e fin
d th
e le
ngth
s of
the
two
legs
of t
he c
reat
ed ri
ght t
riang
le,
then
we
will
hav
e fo
und
the 𝑥
and
𝑦 d
istan
ce, a
nd th
us, t
he 𝑥
- an
d 𝑦
-coo
rdin
ate
Uni
t fiv
e –
146
of o
ur p
oint
? Co
nver
ting
a Po
lar p
oint
to a
rect
angu
lar o
ne, t
hen,
am
ount
s to
fin
ding
leng
ths
of a
tria
ngle
.
But w
e kn
ow h
ow to
do
this!
(A)
Find
the
leng
th o
f eac
h le
g of
the
tria
ngle
sho
wn
abov
e.
(B)
Wha
t, th
en, a
re th
e re
ctan
gula
r coo
rdin
ates
of t
he g
iven
Pol
ar p
oint
?
8.)
Usi
ng th
e sa
me
met
hod
abov
e (a
nd d
raw
ing
a pi
ctur
e), c
onve
rt th
e fo
llow
ing
Pola
r po
ints
into
rect
angu
lar p
oint
s. (A
) 𝐴( 1
,60
°)
(B) 𝐵( 1
,45
°)
(C) 𝐶( 1
,90
°)
(D) 𝐷(2
,𝜋 6)
(E) 𝐸(1
,𝜋 3)
(F)
Did
any
of
your
pre
viou
s re
sults
see
m f
amili
ar t
o w
hat
you’
ve a
lread
y le
arne
d? H
owso
? (G
) Ca
n yo
u ge
nera
lize
the
proc
ess
of c
onve
rtin
g Po
lar
coor
dina
tes
into
re
ctan
gula
r coo
rdin
ates
? It
seem
slik
e th
ere’
sso
me
trea
sure
hid
den
in th
is
Exer
cise
…
§3
The
unit
circ
le
We
now
em
bark
on
perh
aps
the
mos
t im
port
ant s
ectio
n in
Trig
onom
etry
. It i
s im
pera
tive
that
you
lea
rn a
nd m
aste
r th
e te
chni
ques
in
this
sec
tion,
as
it w
ill m
ake
muc
h of
Tr
igon
omet
ry (a
nd, s
ubse
quen
tly, C
alcu
lus)
muc
h ea
sier.
In th
is s
ectio
n w
e pr
ovid
e a
tool
to
hel
p vi
sual
ize
and
effic
ient
ly e
valu
ate
mos
t of
the
Trig
fun
ctio
ns y
ou’ll
run
into
. Of
cour
se, a
s ha
s be
en t
he c
ase
in m
any
of t
hese
Trig
sec
tions
, you
mus
t be
ade
pt a
t th
e pr
evio
us le
sson
s as
wel
l, in
clud
ing
quic
kly
eval
uatin
g si
n3
0°,
for e
xam
ple.
i
Up
to th
is p
oint
, we’
ve w
orke
d w
ith o
nly
a fe
w T
rig fu
nctio
ns, s
uch
as c
os
30
°. An
d, a
s w
e m
entio
ned,
ther
e ar
e m
any
Trig
func
tions
that
we
simpl
y ca
n’t e
valu
ate
with
any
sor
t of
effic
ienc
y. In
this
sect
ion
we’
ll le
arn
how
to e
valu
ate
a ha
ndfu
l mor
e –
an in
finite
am
ount
, ac
tual
ly! –
qui
ckly
and
effi
cien
tly.
Reca
ll th
at th
e m
ain
reas
on w
e’re
abl
e to
eva
luat
e a
Trig
exp
ress
ion
like
tan4
5° i
s bec
ause
w
e ca
n cr
eate
a s
peci
al r
ight
tria
ngle
(in
this
case
, a 4
5°−4
5°−
90
° tr
iang
le) w
hich
we
can
find
the
leng
th o
f th
e sid
es f
or v
ery
quic
kly.
The
n w
e ju
st h
ave
to w
rite
the
i A
lthou
gh, f
rank
ly, a
t thi
s po
int,
this
war
ning
sho
uld
not b
e ne
cess
ary.
§3
The
uni
t circ
le –
147
corr
espo
ndin
g ra
tio.ii
With
out
thes
e sp
ecia
l rig
ht t
riang
les,
we
wou
ld h
ave
to r
esor
t to
ap
prox
imat
ion
met
hods
, whi
ch a
ren’
t ver
y in
tere
stin
g to
stu
dy.
But l
et u
s br
ing
back
our
Pol
ar P
lane
from
the
prev
ious
sec
tion
in F
igur
e 66
. The
n le
t us
see
if w
e ca
n’t c
reat
e an
y ot
her s
peci
al ri
ght t
riang
les.
It tu
rns o
ut th
at n
ot o
nly
can
we
mak
e a
few
, but
we
can
mak
e m
any
of th
em! W
e co
nsid
er
one
exam
ple
belo
w.
Exam
ple
1
Giv
enth
e po
int 𝐴( 2
,12
0°)
, cre
ate
som
e sp
ecia
l rig
ht tr
iang
le.
We
first
plo
t the
giv
en p
oint
. We
show
this
in F
igur
e 67
.
ii W
e m
ust a
lso b
ring
up th
e fa
ct th
at th
e si
ze o
f the
tria
ngle
we
choo
se d
oes
not m
atte
r.
Figu
re 6
6
Can
you
crea
te a
ny s
peci
al ri
ght t
riang
les
usin
g th
is pi
ctur
e? T
ry m
akin
g a
poin
t on
one
of th
e an
gles
.
Uni
t fiv
e –
148
Two
poss
ible
righ
t tria
ngle
pre
sent
them
selv
es, a
nd w
e sh
ow th
em b
oth
in F
igur
e 68
a an
d b.
To d
raw
thes
e rig
ht tr
iang
les,
we
just
dro
pped
a s
trai
ght l
ine
dow
n fro
m th
e po
int t
he 𝑥
-ax
is (in
Fig
ure
a) a
nd th
en a
str
aigh
t lin
e to
the
right
from
the
poin
t to
the 𝑦
-axi
s. Ri
ght
tria
ngle
s ar
e gr
eat,
and
this
allo
ws
us to
use
wha
t we’
ve le
arne
d in
pre
viou
s se
ctio
ns. B
ut
it w
ould
be
even
bet
ter i
f the
y w
ere
spec
ial r
ight
tria
ngle
s, rig
ht?
Figu
re 6
7
Figu
re 6
8a a
nd b
§3
The
uni
t circ
le –
149
But t
hat’s
exa
ctly
wha
t ea
ch o
f the
m a
re! A
nd it
is m
ade
even
mor
e ev
iden
t by
draw
ing
them
on
our P
olar
Pla
ne li
ke w
e di
d. In
Fig
ure
68a,
for e
xam
ple,
you
can
see
that
we
have
a
30
°−
60
°−
90
° tria
ngle
, whi
ch w
e dr
aw s
epar
atel
y in
Fig
ure
69.
This
is s
igni
fican
t, an
d w
e’ll
dem
onst
rate
why
as
we
proc
eed
thro
ugh
this
sect
ion.
Hol
d on
to
this
thou
ght
for
a m
omen
t be
caus
e w
e ne
ed t
o es
tabl
ish s
omet
hing
else
bef
ore
mak
ing
our g
reat
er p
oint
.
One
oft
he E
xerc
ises
from
the
prev
ious
sec
tion
saw
you
con
vert
ing
Pola
r coo
rdin
ates
into
re
ctan
gula
r coo
rdin
ates
. Thi
s was
don
e by
form
ing
a rig
ht tr
iang
le (n
ot u
nlik
e ou
r pre
viou
s Ex
ampl
e) a
nd th
en u
sing
you
r mem
oriz
ed T
rig fu
nctio
ns to
find
the
miss
ing
leng
ths o
f the
rig
ht tr
iang
le (w
hich
cor
resp
onde
d to
the 𝑥
- and
𝑦-v
alue
s of t
he p
oint
iii).
Let u
s rec
onsid
er
that
with
som
e m
ore
form
ality
and
see
if w
e ca
n’t u
ncov
er s
ome
trut
h.
Exam
ple
2a
Conv
ert t
he P
olar
coo
rdin
ate ( 2
,45
°) to
rect
angu
lar c
oord
inat
es.
Alth
ough
you
did
this
prob
lem
in th
e pr
evio
us s
ectio
n, le
t us
form
aliz
e th
is p
roce
ss. W
e fir
st p
lot t
he g
iven
poi
nt, a
nd th
en c
reat
e a
right
tria
ngle
, as
show
n in
Fig
ure
70.
iii H
uh?
Did
you
mis
s so
met
hing
? Th
ere’
s so
met
hing
ver
y in
tere
stin
g go
ing
on h
ere;
can
you
feel
it?
Figu
re 6
9
Uni
t fiv
e –
150
Now
we
can
redr
aw t
hat
tria
ngle
by
itsel
f, ad
ding
in w
hat
we
know
, viz
. the
rad
ius.
We
show
this
in F
igur
e 71
.
Reca
ll th
at w
hen
we
grap
h so
met
hing
on
the
coor
dina
te p
lane
, we
go ri
ght s
ome
dist
ance
, an
d th
en u
p so
me
dist
ance
. In
the
case
of o
ur p
ictu
re, d
o yo
u re
cogn
ize
that
the
orig
in is
th
e po
int t
o th
e le
ft of
the
right
ang
le?
Then
if w
e tr
avel
righ
t to
the
right
ang
le, a
nd th
en
up, w
ould
n’t w
e ha
ve ju
st w
ent t
hrou
gh th
e pr
oces
s of p
lott
ing
a po
int o
n th
e re
ctan
gula
r pl
ane?
iv T
hus,
if w
e fin
d th
e le
ngth
of e
ach
leg,
we’
ll ha
ve o
ur 𝑥
- and
𝑦-c
oord
inat
es, r
ight
?
So th
at’s
wha
t we’
ll do
. And
this
is qu
ite e
asy,
sin
ce th
is is
an is
osce
les
right
tria
ngle
, we
just
div
ide
the
hypo
tenu
se b
y √2
, and
find
that
eac
h le
g ha
s a
leng
th o
f
iv T
hat i
s co
nfus
ing
in w
ords
. Try
doi
ng w
hat I
wro
te to
hel
p yo
u se
e th
at y
ou’re
just
follo
win
g th
e sa
me
proc
edur
e yo
u’ve
use
d pe
rhap
s th
ousa
nds
of ti
mes
to p
lot a
poi
nt. O
r ask
you
r tea
cher
to d
emon
stra
te.
Figu
re 7
0
Figu
re 7
1
§3
The
uni
t circ
le –
151
2 √2
.
Let u
s ra
tiona
lize
this
num
ber s
o w
e ca
n pe
rhap
s re
cogn
ize
it af
ter w
e’re
don
e w
ith it
:
2 √2∙√2
√2
=2√2
2=√2
.
So e
ach
leg
mea
sure
s √2
in le
ngth
, and
this
tells
us
that
the
poin
t 𝐴( 2
,45
°) c
an a
lso b
e w
ritte
n as
𝐴( √2
, √2
).
This
is n
eat,
but m
ore
impo
rtan
tly th
at n
umbe
r √2
rem
inds
us
of a
num
ber t
hat p
oppe
d
up q
uite
a fe
w ti
mes
in th
e pr
evio
us u
nit:
√2 2. I
n fa
ct, o
ur re
sult
was
twic
e th
at o
f √
2 2.
Why
is th
is in
tere
stin
g? B
ecau
se w
hat i
s sin4
5°?
And
wha
t is c
os4
5°?
And
is th
is a
coin
cide
nce?
?
Exam
ple
2b
Conv
ert t
he P
olar
coo
rdin
ate 𝐵( 3
,30
°) in
to re
ctan
gula
r coo
rdin
ates
.
Let’s
do
one
mor
e te
st b
efor
e tr
ying
to g
ener
aliz
e an
d fo
rmal
ize
our r
esul
ts. F
ollo
win
g ou
r pr
evio
us p
roce
dure
, the
n, w
e en
d up
with
Fig
ure
72.
From
her
e, w
e on
ce a
gain
dra
w a
rig
ht t
riang
le, a
nd, o
nce
agai
n, in
Fig
ure
73 w
e se
e a
spec
ial r
ight
tria
ngle
, don
’t w
e?
Figu
re 7
2
Uni
t fiv
e –
152
Agai
n, w
hat w
e’re
look
ing
for a
re th
e le
gs o
f thi
s rig
ht tr
iang
le, a
s th
at w
ill te
ll us
the 𝑥
- an
d 𝑦
-coo
rdin
ates
of t
he p
oint
we’
re lo
okin
g fo
r. U
sing
wha
t we
lear
ned
from
Uni
t fou
r, w
e se
e th
at th
e sh
ort l
eg is
1.5
and
the
long
leg
is 1
.5√
3. A
nd h
ence
our
poi
nt 𝐵
can
be
writ
ten
in th
e re
ctan
gula
r pla
ne a
s 𝐵
(1.5
,1.5√
3).
Let
us a
gain
hig
hlig
ht t
he r
esul
ts: R
emem
ber
that
co
s6
0°
=1 2?
Wel
l, ou
r 𝑥
-coo
rdin
ate
is
that
tim
es 3
, isn
’t it?
Wha
t abo
ut s
in6
0°?
Wha
t rel
atio
nshi
p do
es th
at h
ave
with
our
resu
lt of
1.5√
3?v
Isn’
t it i
nter
estin
g th
at th
e re
ctan
gula
r co
ordi
nate
s of
a p
oint
from
the
Pol
ar P
lane
kee
p co
min
g up
as
mul
tiple
s of
Trig
func
tions
tha
t w
e’ve
mem
oriz
ed?
Let
us n
ow g
ener
aliz
e th
e re
sult
and
mak
e ou
r maj
or p
oint
of t
his
sect
ion.
v Y
ou m
ight
ask
how
we
knew
abo
ut th
is re
latio
nshi
p. G
ood
ques
tion!
We’
re n
ot tr
ying
to te
ach
a pr
oced
ure
here
for y
ou to
follo
w, b
ut o
nly
poin
ting
som
ethi
ng o
ut, v
iz. t
hat t
hese
fam
iliar
num
bers
kee
p po
ppin
g up
. Of c
ours
e, th
ere
is a
reas
on w
e ch
ose
3 in
this
Exa
mpl
e an
d 2
in th
e pr
evio
us…
Can
you
see
w
here
we
get t
hose
two
num
bers
from
?
Figu
re 7
3
§3
The
uni
t circ
le –
153
A po
int o
n a
circ
le in
rect
angu
lar c
oord
inat
es
Any
poin
t on
any
circ
le is
giv
en b
y th
e re
ctan
gula
r coo
rdin
ates
( 𝑟∙c
os𝛼
,𝑟∙s
in𝛼) ,
W
here
𝑟 is
the
radi
us o
f the
circ
le a
nd 𝛼
is th
e an
gle
of ro
tatio
n fro
m th
e po
sitiv
e 𝑥
-axi
s to
the
poin
t.
W
e ca
nnot
ove
rsta
te h
ow im
port
ant t
his
disc
over
y is.
So
let u
s re
stat
e it
anot
her w
ay. W
e ca
n fin
d th
e co
ordi
nate
s of
any
poi
nt o
n a
circ
le u
sing
our
Trig
fun
ctio
ns a
nd t
he
info
rmat
ion
abov
e.
Perh
aps
mor
e im
port
antly
, how
ever
, is
that
this
esta
blis
hes
a re
latio
nshi
p th
at w
e ca
n us
e to
find
the
valu
es o
f ot
her
angl
es t
hat
we
inpu
t in
to T
rig f
unct
ions
. In
our
orig
inal
de
finiti
on o
f Trig
func
tions
, we
coul
d on
ly u
se a
cute
ang
les.
This
rela
tions
hip,
nam
ely
that
𝑥=𝑟∙c
os𝛼
,𝑦=𝑟∙s
in𝛼
,
Whe
re 𝑥
and
𝑦 h
ave
the
usua
l m
eani
ng a
s 𝑥
- an
d 𝑦
-coo
rdin
ates
of
a po
int
in t
he
rect
angu
lar p
lane
, hel
ps u
s to
find
the
outp
ut o
f any
ang
le p
ut in
to a
Trig
func
tion.
With
so
me
sim
ple
Alge
bra,
we’
ll m
ake
it m
ore
clea
r:
cos𝛼
=𝑥 𝑟
,sin𝛼
=𝑦 𝑟
.
Let u
s no
w p
ract
ice
this
conc
ept.
Exam
ple
2a
Eval
uate
sin
12
0°.
Uni
t fiv
e –
154
Did
you
mem
oriz
e si
n12
0°?
Bec
ause
you
shou
ldn’
t hav
e. T
here
’s a
bett
er w
ay to
dea
l with
th
ese
angl
es th
an si
mpl
y m
emor
izin
g.vi U
sing
our
Pol
ar P
lane
from
bef
ore,
we
plac
e so
me
poin
t on
the
12
0° a
ngle
. We
did
this
in E
xam
ple
1, a
nd, s
ince
the
prev
ious
Exa
mpl
e ha
d a
radi
us o
f 2, l
et’s
stic
k w
ith th
at.vi
i
Then
we
can
crea
te th
e rig
ht tr
iang
le w
e sa
w in
Fig
ure
68a,
whi
ch w
e re
draw
in F
igur
e 74
w
ith th
e ra
dius
’ len
gth
adde
d. N
ote
that
bas
ed o
n th
e w
ay w
e ca
me
abou
t our
pre
viou
s de
finiti
on, t
hat i
s, th
at c
os𝛼
=𝑥 𝑟, w
e m
ust u
se th
e 𝑥
-axi
s as
the
base
of o
ur tr
iang
le.vi
ii
Then
, usin
g th
e re
latio
nshi
p pr
evio
usly
sta
ted,
viz
. tha
t
sin𝛼
=𝑦 𝑟
,
we
can
easil
y ge
t ou
r an
swer
. Alth
ough
we
inpu
t 12
0°,
wha
t w
e ar
e w
orki
ng w
ith is
the
6
0° a
ngle
see
n in
the
tria
ngle
in F
igur
e 74
.
We
see
that
𝑟=2
, bec
ause
that
was
our
cho
sen
radi
us. B
ut w
hat i
s 𝑦
? In
this
case
, it’s
the
vert
ical
leg
of t
he t
riang
le d
raw
n in
Fig
ure
74. W
hat
is th
e le
ngth
of t
his
leg?
Hop
eful
ly
you’
ve n
ot f
orgo
tten
abo
ut 3
0°−
60
°−
90
° tr
iang
les,
as w
e’ll
find
the
valu
e of
tha
t le
g us
ing
this
tech
niqu
e. W
e sh
ow th
e le
ngth
s of
the
tria
ngle
in F
igur
e 75
.
vi A
lthou
gh, t
hat b
eing
sai
d, d
on’t
let u
s st
op y
ou fr
om m
emor
izin
g if
that
’s w
hat y
ou’re
real
ly g
ood
at.
vii D
oes
the
radi
us n
eed
to b
e 2
? Ex
celle
nt q
uest
ion!
We’
ll co
ver t
hat i
n Ex
ampl
e 3.
vi
ii By
“bas
e” w
e m
ean
that
the
right
ang
le m
ust b
e lo
cate
d on
the 𝑥
-axi
s. Th
is w
as n
ot th
e ca
se in
Fig
ure
68b.
It is
pos
sible
to u
se th
at F
igur
e, b
ut th
en w
e w
ould
nee
d to
cha
nge
our d
efin
ition
.
Figu
re 7
4
§3
The
uni
t circ
le –
155
Now
we
have
eno
ugh
info
rmat
ion
to w
rite
our a
nsw
er; w
e ge
t
sin
12
0°
=√
3 2.ix
Let’s
do
anot
her e
xam
ple.
Exam
ple
2b
Wha
t is
cos22
5°?
We
follo
w th
e sa
me
proc
edur
e. T
his
time,
we
need
to p
lot a
poi
nt o
n th
e Po
lar P
lane
at
22
5°.
Wha
t rad
ius
shou
ld w
e us
e? H
ow a
bout
we
stic
k w
ith 2
, for
con
sist
ency
’s sa
ke. W
e sh
ow th
is po
int,
that
is, 𝐵( 2
,22
5°)
in F
igur
e 76
.
ix T
hat s
eem
s od
dly
fam
iliar
… W
asn’
t tha
t the
sam
e th
ing
as s
in6
0°…
?
Figu
re 7
5
Uni
t fiv
e –
156
We
now
dra
w a
righ
t tria
ngle
usi
ng th
e 𝑥
-axi
s as
that
bas
e fo
r our
righ
t ang
le. W
e sh
ow
this
in F
igur
e 77
.
Sinc
e w
e’re
look
ing
for C
osin
e, a
nd, b
ased
on
our f
indi
ngs,
cos𝛼
=𝑥 𝑟
,
we
need
to fi
nd th
e le
ngth
of t
he h
oriz
onta
l leg
.
Then
we
just
use
our
kno
wle
dge
of s
peci
al r
ight
tria
ngle
s to
com
plet
e th
e tr
iang
le. W
e ge
t Fig
ure
78, a
s sh
own.
Figu
re 7
6
Figu
re 7
7
§3
The
uni
t circ
le –
157
We
can
now
sub
stitu
te, s
ince
we
know
𝑟 (w
e ch
ose
it to
be 2
) and
𝑥, w
hich
is th
e le
ngth
of
the
horiz
onta
l leg
, whi
ch is
2. W
e ha
ve
cos22
5°
=√2 2
.
But i
s th
is co
rrec
t? O
ne w
ay to
che
ck o
ur re
sult
is to
plo
t it i
n th
e co
ordi
nate
pla
ne. O
ur
𝑥-c
oord
inat
e is
give
n by
the
hor
izon
tal d
istan
ce, w
hich
in t
his
case
cor
resp
onds
to
the
horiz
onta
l leg
in F
igur
e 78
.
Aqu
ick
glan
ce s
how
s th
at w
e m
ust b
e w
rong
. The
poi
nt in
Fig
ure
76 w
as in
Qua
dran
t III,
an
d th
is re
quire
s a
nega
tive 𝑥
-val
ue, r
ight
? W
e ha
ve a
n iss
ue.
Or d
o w
e? T
here
is a
n ea
sy w
ay to
rect
ify th
is e
rror
: W
e si
mpl
y m
ake
our r
esul
t neg
ativ
e.
This
seem
s aw
fully
art
ifici
al, a
nd it
is. B
ut th
at’s
whe
re o
ur P
olar
Pla
ne c
omes
in h
andy
. We
have
a p
ictu
re to
see
that
, cle
arly
, our
𝑥-c
oord
inat
e m
ust b
e ne
gativ
e. T
here
fore
, our
resu
lt is
cos22
5°
=−√2 2
.
Let’s
do
one
mor
e ex
ampl
e be
fore
intr
oduc
ing
you
to a
spe
cial
circ
le.
Exam
ple
2c
Eval
uate
all
six T
rig fu
nctio
ns u
sing 5𝜋 3 a
s th
e in
put.
We
are
look
ing
for s
in5𝜋 3
,co
s5𝜋 3
, and
so
on. W
e ha
ve ra
dian
s as
our
inpu
t, an
d th
is is
fine,
since
we
know
how
to
wor
k w
ith t
hem
. The
firs
t th
ing
we
shou
ld d
o is
plot
the
poi
nt
𝐶(2
,5𝜋 3)
on th
e Po
lar P
lane
. We
do th
is in
Fig
ure
79.
Figu
re 7
8
Uni
t fiv
e –
158
Befo
re p
roce
edin
g, d
o yo
u se
e ho
w 𝐶
is in
Qua
dran
t IV?
Thu
s 𝑥
>0
and
𝑦<
0. Y
ou m
ight
w
ant t
o m
ake
a no
te o
f thi
s fo
r eac
h of
the
prob
lem
s lik
e th
is y
ou d
o, s
o th
at y
ou d
on’t
forg
et.
Now
we
crea
te a
righ
t tria
ngle
with
the 𝑥
-axi
s as
our
bas
e. W
e ge
t Fig
ure
80 a
s sh
own.
Now
all
we
need
to d
o is
writ
e th
e ap
prop
riate
ratio
s. W
e se
e th
at
sin
5𝜋 3
=−√
3 2
Figu
re 7
9
Do
you
see
that
5𝜋 3 is
equ
ival
ent t
o 3
00
°?
Figu
re 8
0
Verif
y th
at w
e’ve
com
plet
ed th
is sp
ecia
l rig
ht tr
iang
le c
orre
ctly
.
§3
The
uni
t circ
le –
159
beca
use
Sine
is o
ppos
ite to
hyp
oten
use.
It m
ust b
e ne
gativ
e sin
ce w
e ar
e in
Qua
dran
t IV.
W
e fu
rthe
r see
that
cos
5𝜋 3
=1 2
,
and,
sin
ce w
e ar
e in
Qua
dran
t IV,
we
mus
t be
posit
ive.
Wha
t ab
out
the
othe
r Tr
ig f
unct
ions
? Fi
rst,
reca
ll th
at t
an𝛼
=sin𝛼
cos𝛼. A
nd s
ince
sin
5𝜋 3 is
nega
tive
whi
le c
os5𝜋 3 is
pos
itive
, the
n Ta
ngen
t m
ust
be n
egat
ive.
x The
n w
e ju
st n
eed
to
find
the
ratio
from
the
prev
ious
Fig
ure,
and
mak
e it
nega
tive.
We
get
tan
5𝜋 3
=−√
3.
Find
ing
the
reci
proc
al T
rig f
unct
ions
, lik
e Se
cant
, is
eas
y. W
e ju
st n
eed
to f
ind
the
reci
proc
al o
f the
pre
viou
s th
ree
resu
lts. N
ote
that
find
ing
a re
cipr
ocal
doe
s no
t cha
nge
its
sign.
The
refo
re,
csc
5𝜋 3
=−2 √3
,se
c5𝜋 3
=2
,co
t5𝜋 3
=−
1 √3
.
We
leav
e th
e ra
tiona
lizat
ion
to th
e re
ader
, if t
hey
wis
h.
Up
to th
is po
int,
we’
ve a
lway
s us
ed a
radi
us o
f 2. J
ust b
ecau
se. B
ut c
erta
inly
ther
e ha
s to
be
a b
ette
r cho
ice,
righ
t? In
deed
, the
re is
, and
we
call
it th
e un
it c
ircl
e.
The
unit
circ
le
A ci
rcle
cen
tere
d at
the
o rig
in w
hose
radi
us is
1.
This
sim
ple
defin
ition
has
som
e pr
ofou
nd im
plic
atio
ns. F
or e
xam
ple,
rec
all t
hat,
usin
g a
circ
le o
f rad
ius 𝑟,
we
have
the
rela
tions
hip co
s𝛼
=𝑥 𝑟
.
But i
f we
have
a u
nit c
ircle
, whe
re 𝑟
=1
, the
n w
e ha
ve th
e sim
pler
rela
tions
hip
cos𝛼
=𝑥
.
This
also
hol
ds w
ith s
in𝛼
, of c
ours
e:
sin𝛼
=𝑦
.
x T
he q
uotie
nt o
f a n
egat
ive
num
ber a
nd a
pos
itive
mus
t be
a ne
gativ
e nu
mbe
r, rig
ht?
Uni
t fiv
e –
160
Not
onl
y is
this
easi
er to
writ
e, b
ut it
also
hel
ps u
s to
find
the
Sine
(or C
osin
e, o
r…) o
f any
an
gle
pret
ty e
asily
. We
show
this
intu
ition
in F
igur
e 81
.
Exam
ple
3a
Eval
uate
sin
13
5°.
The
unit
circ
le is
a p
ictu
re to
pla
ce in
to y
our h
ead
so y
ou c
an e
valu
ate
Trig
func
tions
like
th
is ve
ry q
uick
ly, e
ffici
ently
, and
acc
urat
ely.
At f
irst,
you’
ll ne
ed to
dra
w it
out
and
it m
ight
ta
ke s
ome
time.
But
eve
ntua
lly i
t be
com
es s
econ
d-na
ture
, an
d it
is in
disp
ensa
ble
in
Calc
ulus
. So
whi
le s
omeo
ne a
dept
at m
ath
mig
ht n
ot n
eed
to d
raw
it o
ut, w
e w
ill d
o so
in
each
of t
hese
exa
mpl
es.
We
draw
a 1
35
° ang
le o
n th
e un
it ci
rcle
in F
igur
e 82
.
Sinc
e 𝑃
has
𝑥-
and 𝑦
-coo
rdin
ates
of
cos𝛼
and
sin𝛼
, res
pect
ivel
y, a
ll w
e ne
ed t
o do
to
eval
uate
sin
13
5° i
s to
find
the 𝑦
-coo
rdin
ate
of 𝑃
. But
this
amou
nts
to th
e sa
me
thin
g w
e in
the
pre
viou
s se
t of
Exa
mpl
es, u
sing
the
Pola
r Pl
ane.
The
onl
y di
ffere
nce
is t
hat
our
Figu
re 8
1
Any
poin
t on
the
circ
le h
as a
n 𝑥
-coo
rdin
ate
of c
os𝛼
and
a 𝑦
-coo
rdia
nte
of s
in𝛼
.
Figu
re 8
2
§3
The
uni
t circ
le –
161
radi
us w
ill a
lway
s be
1.xi
Thu
s we
need
to m
ake
a rig
ht tr
iang
le w
ith th
e 𝑥
-axi
s as o
ur b
ase
as s
how
n in
Fig
ure
83.
Now
we
just
use
sub
stitu
te. W
e ge
t tha
t
sin
13
5°
=√2 2
.
And
yes,
our r
esul
t sho
uld
be p
ositi
ve. A
noth
er w
ay to
see
that
our
resu
lt m
ust b
e po
sitiv
e is
beca
use,
if w
e st
art a
t the
orig
in, w
e ha
d to
trav
el u
p to
get
to 𝑃
, cor
rect
? An
d isn
’t up
a
posit
ive
dire
ctio
n?
Exam
ple
3b
Eval
uate
co
s11𝜋
6.
We
first
dra
w a
n an
gle
of 11𝜋
6 o
n ou
r uni
t circ
le in
Fig
ure
84.
xi A
s op
pose
d to
wha
teve
r we
wan
t. W
hy c
hoos
e 1
, aga
in?
Figu
re 8
3
If yo
u ha
ve d
iffic
ultie
s se
eing
wha
t typ
e of
spe
cial
righ
t tria
ngle
you
get
, use
you
r Pol
ar P
lane
. It w
ill b
e m
ore
evid
ent s
ince
ea
ch a
ngle
is m
arke
d. A
lso n
ote
we
ratio
naliz
ed th
e de
nom
inat
ors
on e
ach
leg.
Uni
t fiv
e –
162
Agai
n, w
eno
w c
reat
e a
right
tria
ngle
usin
g th
e 𝑥
-axi
s as o
ur b
ase.
We
get t
he sp
ecia
l rig
ht
tria
ngle
sho
wn
in F
igur
e 85
.
Then
we
just
sub
stitu
te, k
now
ing
wha
t rat
io w
e ge
t with
the
Cosin
e fu
nctio
n. H
ence
cos
11𝜋
6=
1 2.
This,
also
, sho
uld
be p
ositi
ve, s
ince
we
wen
t to
the
right
to g
et to
𝑃.
Be p
atie
nt a
nd r
esili
ent
as y
ou l
earn
the
uni
t ci
rcle
. Mas
tery
will
com
e, b
ut o
nly
with
pr
actic
e an
d pe
rsev
eran
ce. O
nce
you
mas
ter t
he u
nit c
ircle
, Trig
onom
etry
bec
omes
you
r pl
ayth
ing.
§𝟑 E
xerc
ises
1.)
Plot
the
follo
win
g po
ints
on
the
Pola
r Pla
ne, t
hen
crea
te a
righ
t tria
ngle
whe
re th
e 𝑥
-axi
s se
rves
as
the
base
. (A
) 𝐴( 2
,13
5°)
(B
) 𝐵( 3
,30
0°)
(D
) 𝐷(3
,𝜋 6)
Figu
re 8
4
Agai
n, u
se y
our P
olar
Pla
ne to
hel
p yo
u fin
d 11𝜋
6. H
owev
er, y
ou’ll
wan
t to
have
a v
ery
firm
gra
sp o
f rad
ians
so
don’
t
com
plet
ely
rely
on
you
Pola
r Pla
ne.
Figu
re 8
5
Remote Learning Packet Please submit scans of written work in Google Classroom at the end of the week.
Week 7: May 11-15, 2020 Course: Spanish III Teacher(s): Ms. Barrera [email protected] Supplemental links: www.spanishdict.com www.lingt.com/barreratumble Weekly Plan: Monday, May 11 ⬜ Capítulo 4B - Part I Read about the life and customs of Spain. ⬜ Capítulo 4B - Reading comprehension of vocabulary classification and matching. Tuesday, May 12 ⬜ Capítulo 4B - Part II Read about the life and customs of Spain. ⬜ Capítulo 4B - Reading comprehension by completing the statements and definition of festivals. Wednesday, May 13 ⬜ Capítulo 4B- Listening Activity: Spanish storytime from the book titled Vida o muerte en el Cusco. ⬜ Capítulo 4B- Speaking Activity: Listen and answer questions in Spanish. Thursday, May 14 ⬜ Capítulo 4B- Painting by Carmen Lomas Garza, Tamalada, 1990, color lithograph, Smithsonian American Art Museum. ⬜ Capítulo 4B- Read statements about the painting and decide whether true or false and correct if false. Friday, May 15 ⬜ attend office hours ⬜ catch-up or review the week’s work Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature
I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature
Monday, May 11 Capítulo 4B - Part I - Read about the life and customs of Spain. Comprehension of the reading using matching, vocabulary classification and matching. I. Reading Handout: La vida y las costumbres de España. Reading comprehension. Exercise A. Indicate for each of the 15 words if they belong under alimentation, beverages, garments, religious festivals or social customs. Exercise B. Match the 10 vocabulary words with the words on the right side of the column. Tuesday, May 12 Capítulo 4B - Part II - Read about the life and customs of Spain. Comprehension of the reading by completing the statements and definition of festivals. I. Reading Handout: La vida y las costumbres de España. Reading comprehension. Exercise C. Complete the following 10 sentences with the appropriate vocabulary. Exercise D. Explain each of these 10 religious holidays. For example; romería Cada pueblo tiene su santo patrón, en el dia del santo se va a la tumba del santo a hacer picnics religiosas. Esos picnics se llaman romerías. Wednesday, May 13 Capítulo 4B- Listening Activity: Storytime in Spanish from the book titled Vida o muerte en el Cusco. Speaking Activity: Listen and answer questions in Spanish. I. Storytime: Listen to chapters 2 and 3 of Vida o muerte en el Cusco. Video link in google classroom. II. Speaking Assignment : Answer questions in Spanish about chapter 2 and 3. Questions are in lingt. Thursday, May 14 Capítulo 4B- Painting by Carmen Lomas Garza, Tamalada, 1990, color lithograph, Smithsonian American Art Museum. Read statements about the painting and decide whether true or false and correct if false. I.Textbook, p.222 - Activity 13 - La Tamalada, Painting by Carmen Lomas Garza. Read the following statements and decide if they are true or false. If they are false please correct and write the entire sentence. These sentences are in the preterit and imperfect tense.
1. Entre en la cocina con mi papa. 2. Había tres personas alrededor de la mesa. 3. Todos ayudaban a hacer galletas. 4. Todos mis parientes tenían el pelo negro. 5. Las personas tenían diferentes trabajos. 6. Las paredes de la cocina eran amarillas.
