date: 4.1 radian and degree measuremeservey.pbworks.com/w/file/fetch/48434397/precalc...page 7 of 24...
TRANSCRIPT
Page 1 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.1 Radian and Degree Measure
Syllabus Objective: 3.1 – The student will solve problems using the unit circle.
Trigonometry means the measure of triangles.
Standard Position (of an angle): initial side is on the positive x-axis;
positive angles rotate counter-clockwise
negative angles rotate clockwise
Radian: a measure of length; 1 radian when r s
r = radius, s = length of arc
s
r
Arc Length: s r ( must be measured in radians)
Recall: Circumference of a Circle; 2C r
So, there are 2 radians around the circle 2 radians = 360°, or radians = 180
Draw in the radians: Degrees and Radians:
Coterminal Angles: angles with the same initial and terminal sides, for example 360 or 2
Ex1: Find a positive and negative coterminal angle for each.
a. 130
b. 9
2
Initial side
Terminal side
r θ
s
Page 2 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Complementary Angles: The sum of two angles is 2
or 90
Supplementary Angles: The sum of two angles is or 180
Ex2: If possible find the complement and the supplement of
a.) 2
5
b.) 4
5
Converting Degrees (D) to Radians (R): 180
D R
(Note: 1180
)
Ex3: Convert 540° to radians.
Note: We multiply by 180
so that the degrees cancel. This will help you remember what to
multiply by.
Converting from Radians (R) to Degrees (D): 180
D R
Or use the proportion: 180
d r
Ex4: Convert 3
4
radians to degrees.
Note: We multiply by 180
so that the ’s cancel.
Page 3 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Reference Angle: every angle has a reference angle, with initial side as the x-axis
Ex.5
Draw and Find the reference angle of a.) 45 degrees, b.) 120 degrees, c.) 225 degrees, d.) 330 degrees.
Special Angles to Memorize (Teacher Note: Have students fill this in for practice.)
Degrees 30° 45° 60° 90° 120° 135° 150° 180°
Radians 6
4
3
2
2
3
3
4
5
6
Shortcut for Specials:
Degrees to radians
1. Find the reference angle
2. Divide reference angle into angle
3. Write reference angle in radians times answer in #2. 33011
330 30 times 116 6
Radians to degrees:
1. Write reference angle
2. Multiply example 4
4 times 60 2403
Reflection:
Date: 4.1 Radian and Degree Measure Continued Review:
Convert 1.2 hours into hours and minutes.
Solution:
Convert 3 hours and 20 minutes into hours.
Solution:
Babylonian Number System: based on the number 60; 360 approximates the number of days in a year.
Circles were divided into 360 degrees.
A degree can further be divided into 60 minutes (60'), and each minute can be divided into 60 seconds
(60'').
Page 4 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Converting from Degrees to DMS (Degrees – Minutes – Seconds): multiply by 60
Ex1: Convert 114.59° to DMS.
Converting to Degrees (decimal): divide by 60
Ex2: Convert 72° 13' 52'' to decimal degrees.
Note: The degree and minute symbols can be found in the ANGLE menu. The seconds symbol can be
found above the + sign (alpha +).
Arc Length: s r ( must be measured in radians)
Ex3: A circle has an 8 inch diameter. Find the length of an arc intercepted by a 240° central
angle.
Step One: Convert to radians.
Step Two: Find the radius.
Step Three: Solve for s.
Linear Speed (example: miles per hour): length
time
s rl
t t
Angular Speed (example: rotations per minute): angle
timeA
t
Ex4: Find the linear and angular speed (per second) of a 10.2 cm second hand.
Note: Each revolution generates 2 radians.
a.) Linear Speed: A second hand travels half the circumference in ____ seconds: or
b.) Angular Speed:
Note: The angular speed does not depend on the length of the second hand!
For example: The riders on a carousel all have the same angular speed yet the riders on the outside have a
greater linear speed than those on the inside due to a larger radius.
Page 5 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex5: Find the speed in mph of 36 in diameter wheels moving at 630 rpm (revolutions per
minute).
Unit Conversion:
1 statute (land) mile = 5280 feet Earth’s radius 3956 miles
1 nautical mile = 1 minute of arc length along the Earth’s equator
Ex6: How many statute miles are there in a nautical mile?
s r 3956r
Bearing: the course of an object given as the angle measured clockwise from due north
Ex7: Use the picture to find the bearing of the ship.
?
You Try
1. A lawn roller with 10 inch radius wheels makes 1.2 revolutions/second. Find the linear and
angular speed.
2. How many nautical miles are in a statute mile? Show your work.
Reflection:
QOD: How are radian and degree measures different? How are they similar?
N
51°
51°
N
Page 6 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.2 Trigonometric Functions: The Unit Circle
Syllabus Objectives: Syllabus Objectives: 3.1 – The student will solve problems using the unit
circle. 3.2 – The student will solve problems using the inverse of trigonometric functions.
The coordinates x and y are two functions of the real variable t. You can use these coordinates to define
the six trigonometric functions of t.
sine cosecant
cosine secant
tangent cotangent
Let t be a real number and let (x,y) be the point on the unit circle corresponding to t.
1sin csc
1cos sec =
tan cot
t y ty
t x tx
y xt t
x y
The Unit Circle: 1r
Page 7 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex 1: Evaluate the six trigonometric functions at each real number.
a.) 6
t
b.) 5
4t
c.) 0t d.) t
You Try: Find the following without a calculator.
1. os
2. 5
cot3
3. sin225
Periodic Functions: A function y f t is periodic if there is a positive number c such that
f t c f t for all values of t in the domain of f.
The period of sine and cosine are 2 and the period of tangent is .
Ex2: Evaluate 3601
sin2
without a calculator.
Exploration: Consider an angle, θ, and its opposite, as shown in the coordinate grid. Compare the trig
functions of each angle.
sin , sin cos , cos tan , tan
csc , csc sec , sec cot , cot
**Cosine and secant are the only EVEN f x f x trig functions.
All the rest are ODD f x f x .
Sin t
Cos t
Tan t
Csc t
Sec t
Cot t
x
z
z
y
-y
θ −θ
Page 8 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Odd-Even Identities:
Ex3: Find a.) sin (-t) b.) cos (-t)
c.) Simplify the expression sin cscx x .
Evaluating Trigonometric Ratios on the Calculator
Note: Check the MODE on your calculator and be sure it is correct for the question asked (radian/degree).
Unless an angle measure is shown with the degree symbol, assume the angle is in radians.
Ex5: Evaluate the following using a calculator.
a. sin42.68 Mode: degree
sin42.68 0.678
b. sec1.2 Mode: radian
Note: There is not a key for secant. We must use 1/(cosx) because secant is the reciprocal of
cosine. sec1.2 2.760
Also recall: 1 1
csc and cotsin tan
x xx x
c. tan72 13 52 Mode: degree
Inverse Trigonometric Functions: use these to find the angle when given a trig ratio
Ex6: a.) Find θ (in degrees) if 5
cos12
.
1 5cos
12
Mode: degrees 65.376
b.) sin t = 1/3
REFLECTION: How is the unit circle used to evaluate the trigonometric functions? Explain.
sin sin and csc csc
tan tan and cot cot
cos cos and sec sec
Page 9 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.3 Right Triangle Trigonometry
Syllabus Objectives: 3.1 – The student will solve problems using the unit circle. 3.2 – The student
will solve problems using the inverse of trigonometric functions. 3.5 – The student will solve
application problems involving triangles.
Note: The adjacent and opposite sides are always the legs of the right triangle, and depend upon which
angle is used.
Trigonometric Ratios of θ: Reciprocal Functions
Sine: opposite
sinhypotenuse
Cosecant: hypotenuse
cscopposite
Cosine: adjacent
coshypotenuse
Secant: hypotenuse
secadjacent
Tangent: opposite
tanadjacent
Cotangent: adjacent
cotopposite
Memory Aid: SOHCAHTOA
Think About It: Which trigonometric ratios in a triangle must always be less than 1? Why?
Ex1: For ABC , find the six trig ratios of A .
sin A cos A tan A
csc A sec A cot A
Note: It may help to label the sides as Opp, Adj, and Hyp first.
Find the six trig ratios for B .
sin A cos A tan A
csc A sec A cot A
Opposite
Adjacent Hypotenuse
10
8
6
B
A C
θ
Adjacent
Opposite Hypotenuse
Page 10 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Special Right Triangles: 30-60-90 & 45-45-90
Formulate Chart. This table must be memorized!
θ (Degrees) 30° 45° 60°
θ (Radians)
0 6
4
3
2
sinθ
cosθ
tanθ
4.3 Identities
Special Relationships: Identity: a statement that is true for all values for which both sides are defined
Example from algebra: 3 8 11 3 13x x
Exploration: Consider a right triangle.
Note that and are complementary. Write the trig functions for each angle. What do you notice?
sin cos cos sin tan cot
csc sec sec csc cot tan
**The trig functions of are equal to the cofunctions of θ, when and are complementary.
a
b
c
θ
Φ
x 3
30
°
x
2
x
45°
x
x 2
x
3x7
Page 11 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Cofunction Identities:
Ex2. a. sin 1 ˚=
. tan =
c. sec 25˚=
Ex3. cos α = .8 Find sin α and tan α using identities.
Reciprocal Identities
1 1 1sin cos tan
csc sec cot
1 1 1csc sec cot
sin cos tan
Quotient Identities sin cos
tan cotcos sin
Recall: Unit Circle 1, cos , sinr x y
Pythagorean Theorem: 2 2 1x y Pythagorean Identity: 2 2sin cos 1
To derive the other Pythagorean Identities, divide the entire equation by 2sin and then by 2cos :
sin 90 cos or sin cos2
cos 90 sin or cos sin2
sec 90 csc or sec csc2
csc 90 sec or csc sec2
tan 90 cot or cot tan2
cot 90 tan or cot tan2
,x y
Note: 22sin sin
Page 12 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Pythagorean Identities 2 2 2 2 2 2sin cos 1 1 cot csc tan 1 sec
Simplifying Trigonometric Expressions:
Look for identities
Change everything to sine and cosine and reduce
Ex4: Use basic identities to simplify the expressions.
a) 2cot 1 cos cos
cotsin
2 2 2 2
sin cos 1 sin 1 cos
b) tan csc sin
tancos
1csc
sin
Ex5: Simplify the expression
2
csc 1 csc 1
cos
x x
x
.
Use algebra: 2 2 2 21 cot csc cot csc 1
Application Problem
Ex6: A 6 ft 2 in man looks up at a 37 angle to the top of a building. He places his heel to toe 53
times and his shoe is 13 in. How tall is the building?
Draw a picture: The height of the building is (y + 74) inches.
Use trig in the right triangle to solve for y:
Height of the building in inches =
Convert to feet: The building is approximately _____________________ tall.
74''
53(13)=689''
y
37°
74''
Page 13 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
You Try: Solve the triangle (find all missing sides and angles).
Reflection
QOD: Explain how to find the inverse cotangent of an angle on the calculator.
M
N
Q
q m
8
32°
Page 14 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.4 Trigonometric Functions of Any Angle
Syllabus Objectives: 3.1 – The student will solve problems using the unit circle. 3.2 – The student
will solve problems using the inverse of trigonometric functions.
Trigonometric Functions of any Angle
Review: Memory Aid: To remember which trig functions are positive in which quadrant, remember
All Students Take Calculus. A – all are positive in QI, S – sine (and cosecant) is positive in QII, T –
tangent (and cotangent) is positive in QIII, C – cosine (and secant) is positive in QIV.
Review: Reference Angle: every angle has a reference angle, with initial side as the x-axis
Have Memorized:
Degrees 0°/360° 30° 45° 60° 90° 180° 270°
Radians 0 / 2
6
4
3
2
3
2
sinθ 0 1
2
2
2
3
2 1 0 −1
cosθ 1 3
2
2
2
1
2 0 −1 0
tanθ 0 3
3 1 3 und 0 und
1. Find the Quadrant (Use: All Students Take Calculus)
2. Find the Sign (positive or Negative)
3. Find Reference Angle
4. Evaluate (see chart above)
*See timed test of Trigonometric Functions.
2 2 2 2 2x y r r x y
sin csc
cos sec
tan cot
y r
r y
x r
r x
y x
x y
r
II (S)
sin (+)
tan (+)
III (T)
cos (+)
IV (C)
I (A)
sin (+)
cos (+)
tan (+) x
y
θ
Page 15 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex1: Find the six trig functions if 5
sin7
and tan 0 .
Since both sine and tangent are positive, we know that θ is in the ________ quadrant. Find the missing
leg (a): 2 2 25 7 _____________a a This is the adjacent leg of angle α.
5 oppsin
7 hyp
y
r
sin csc
cos sec
tan cot
Ex2: Find the sin of 8
3
Solving a Triangle: find the missing angles and sides with given information
Ex3: Solve the triangle (find all missing sides and angles).
4a (Pythagorean Triple: 3-4-5; or use the Pythagorean Thm)
5 7
α
A
B C
3
a
5
Page 16 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Evaluating Trig Functions Given a Point
Use the ordered pair as x and y
Find r: 2 2r x y
Check the signs of your answers by the quadrant
Ex4: Find the six trig functions of θ in standard position whose terminal side contains
point 5,3 .
Note: The point is in QII, so sine (and cosecant) will be positive.
sin csc
cos sec
tan cot
Ex5: Find the 6 trig functions of 330 .
To find the reference angle, start at the positive x-axis and go counter-clockwise 330 .
Reference Angle =____ ______, _____, ______x y r (30-60-90)
sin csc
cos sec
tan cot
Quadrantal Angles: angles with the terminal side on the axes, for example, 0 ,90 ,180 ,270
Needs to be memorized!
Degrees 0°/360° 30° 45° 60° 90° 180° 270°
Radians 0 / 2
6
4
3
2
3
2
sinθ
cosθ
tanθ
Ex6: Find csc13 .
Find the reference angle:
Reflection:
3
1
Page 17 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.5 Graphs of Sine and Cosine Functions
Syllabus Objectives: 4.1 – The student will sketch the graphs of the six trigonometric functions. 4.3
– The student will graph transformations of the basic trigonometric functions. 4.6 – The student
will model and solve real-world application problems involving sinusoidal functions.
Teacher Note: Have students fill out the table as quickly as they can. Discuss patterns they can use to be
able to memorize these.
θ 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 360°
radians 0 6
4
3
2
2
3
3
4
5
6
7
6
5
4
4
3
3
2
2
sinθ
cosθ
tanθ
Sinusoid: a function whose graph is a sine or cosine function; can be written in the form
siny a bx c d
Sinusoidal Axis: the horizontal line that passes through the middle of a sinusoid
EX1 Graph of the Sine Function
Use the table above to sketch the graph.
a.) Radians: siny x b.) Degrees: siny
Sine is periodic, so we can extend the graph to the left and right.
Characteristics:
Domain: Range: y-intercept: x-intercepts:;
Absolute Max = Absolute Min = Decreasing:
Increasing: Period = Sinusoidal Axis:
Page 18 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
EX 2 Graph of the Cosine Function
Use the table above to sketch the graph.
a.) Radians: cosy x b.) Degrees: cosy
Cosine is periodic, so we can extend the graph to the left and right.
Characteristics:
Domain: Range: y-intercept:
x-intercepts: Absolute Max = Absolute Min =
Decreasing: Increasing: Period = Sinusoidal Axis:
Note: We will work mostly with the graphs in radians, since it is the graph of the function in terms of x.
Recall: 2
f x a x h k is a transformation of the graph of 2f x x . a represents the vertical
stretch/shrink, h is a horizontal shift and k is a vertical shift.
Transformations of Sinusoids:
General Form siny a b x h k
a: vertical stretch and/or reflection over x-axis amplitude = a
h: horizontal shift phase shift = h k: vertical shift
period = 2
b
frequency =
2
b
b = number of cycles completed in 2
Amplitude: the distance from the sinusoidal axis to the maximum value (half the height of a wave)
Ex3: Sketch the graph of 4siny x .
Vertical stretch: Amplitude =
Page 19 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Period: the length of time taken for one full cycle of the wave 2
Pb
Frequency: the number of complete cycles the wave completes per unit of time freq = 2
b
Ex4: Sketch the graph of sin 2y x .
Period = Frequency =
Reflection: If 0a , the sinusoid is reflected over the _____axis.
Ex5: Sketch the graph of cosy x .
Note: Period, amplitude, etc. all stay the same.
Vertical Translation: in the sinusoid siny a b x h k , the line y k is the sinusoidal axis
Ex6: Sketch the graph of 4 siny x . Sinusoidal Axis:
Phase Shift: the horizontal translation, h, of a sinusoid siny a b x h k
Page 20 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex7: Sketch the graph of
cos2
y x
.
Shift cosy x to the right 2
. Does this graph look familiar? It is _______________________
Note: Every cosine function can be written as a sine function using a phase shift.
cos sin2
y x y x
Ex8: Sketch the graph of cos 4y x . Then rewrite the function as a sine function.
The coefficient of x must equal 1. Factor out any other coefficient to find the actual horizontal shift.
cos 4 cos 44
y x y x
Phase Shift: Period:
Graph:
Write as a sine function: cos 4y x
Writing the Equation of a Sinusoid
Ex9: Write the equation of a sinusoid with amplitude 4 and period 3
that passes through 6,0 .
Amplitude: ; Period: 2
3b
b
;
Phase Shift: normally passes through 0,0 , so shift right 6 units
Page 21 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
You Try: Describe the transformations of 7sin 2 0.54
y x
. Then sketch the graph.
Reflection
QOD: How do you convert from a cosine function to a sine function? Explain.
Page 22 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Date: 4.6 Graphs of Other Trigonmetric Functions
Syllabus Objectives: 4.1 – The student will sketch the graphs of the six trigonometric functions. 4.3
– The student will graph transformations of the basic trigonometric functions.
θ 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 360°
radians 0 6
4
3
2
2
3
3
4
5
6
7
6
5
4
4
3
3
2
2
sinθ
cosθ
tanθ
Ex 1 Graph of the Tangent Function: Use the table above to sketch the graph.
a.) Radians: tany x b) Degrees: tany
Characteristics:
Domain: Range: Intercepts:
Increasing/Decreasing:
Period: Vertical Asymptotes:
Note: sin
tancos
xx
x , so the zeros of sine are the zeros of tangent, and the zeros of cosine are the vertical
asymptotes of tangent.
Ex 2 Graph of the Cotangent Function
Use the table above to sketch the graph. (Cotangent is the reciprocal of tangent.)
a.) Radians: coty x b.) Degrees: coty
Characteristics:
Domain: Range: Intercepts:
Increasing/Decreasing Period: Vertical Asymptotes:
Note: cos
cotsin
xx
x , so the zeros of cosine are the zeros of cotangent, and the zeros of sine are the vertical
asymptotes of cotangent.
Page 23 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex. 3 Graph of the Secant Function
Use the table above to sketch the graph. (Secant is the reciprocal of cosine.)
a.) Radians: secy x b.) Degrees: secy
Note: It may help to graph the cosine function first.
Characteristics:
Domain: Range: Intercept:
Local Max: Local Min: Period: Vertical Asymptotes:
Ex. 4 Graph of the Cosecant Function
Use the table above to sketch the graph. (Cosecant is the reciprocal of sine.)
a.) Radians: cscy x b.) Degrees: cscy
Note: It may help to graph the sine function first.
Characteristics:
Domain: Range: Intercept:
Local Max: Local Min: Period: Vertical Asymptotes:
Transformations: Sketch the graph of the reciprocal function first!
Ex5: Sketch the graph of 1 2secy x .
Graph 1 2cosy x . Shift up 1, amplitude 2. Use the zeros of the cosine function to determine the
vertical asymptotes of the secant function. Sketch the secant function.
Note: Amplitude is not applicable to secant.
The number 2 represents a vertical stretch.
Page 24 of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6
Ex6: Sketch the graph of tan2
xy
.
Period: 21
2
P
Reflect over x-axis
Solving a Trigonometric Equation Algebraically
Ex7: Solve for x in the given interval using reference triangles in the proper quadrant.
sec 2,2
x x
Quadrant II; cos x Reference Angle = In QII:
Solving a Trigonometric Equation Graphically
Ex8: Use a calculator to solve for x in the given interval: cot 5, 02
x x
The equation cot 5x is equivalent to the equation 1
tan5
x . Graph each side of the equation and find
the point of intersection for 02
x
. Restrict the window to only include these values of x.
Note: We could have also graphed 1 2
1and 5
tany y
x . (Shown in the second graph above.)
Solution: _____________________________
You Try: Graph the function csc 2y x .
Reflection: QOD:
1. What trigonometric function is the slope of the terminal side of an angle in standard position?
Explain your answer using the unit circle.
2. Which trigonometric function(s) are odd? Which are even?