9/29 test wednesday pick up review & calculator have motion ws ii out warm up 2: look at the...

9/29

Test Wednesday

Pick up review & calculator

Have Motion WS II out

Warm up 2: Look at the V v T graph below. Draw the D v T graph showing the same motion.

V v T

m/s

sec

Make sure you write date, question, & answer for warm ups. If you are absent you must get them from the back.

10/1

Pick up calculator and trig notes. Make sure calculator is in degree mode Tue you handed in Motion WS II 1-12 & worked on

Motion Review Wed you took the Motion Test Test corrections/retakes: Thur pm, Mon am & pm,

Tue am ONLY Warm Up #3 List 3 facts about triangles:

10/5

Today you will finish Trig Packet Tomorrow you will have a quiz No warm up today I will be here after school 11i) 0.633

10/6 YESTERDAY YOU FINISHED TRIG PACKET. THERE WAS NOT A WARM UP. QUIZ TODAY

Warm Up 4: Sketch the triangle

We will solve together

What do the 2 triangle have in common

Solve that first

12.68 cm

Now solve for X

12.98º 25º

30cm

55 cm

X

10/6 YESTERDAY YOU FINISHED TRIG PACKET. THERE WAS NOT A WARM UP. QUIZ TODAY

Warm Up 4: Solve for X

25º

30cm

55 cm

X

10/7 YESTERDAY YOU HAD A QUIZ. QUIZ CAN BE MADE UP AFTER SCHOOL TODAY OR BEFORE SCHOOL ON THURSDAY

Pick up Homework Set (Empire State) This is due BOC Thursday

Today we will be going outside to do the survey lab. (After we finish ex F of the application notes)

NOW: Have your notes out.

Trig Test will be Friday Oct 16

Warm Up #5

What is the missing angle?

What can we say about sides A & B?

TURN IN WARM UPS TO SORTER

45º

XB

A

ANSWERS TO EMPIRE STATE PROBLEM SET

1. 36.45º 28.07º 2. 335.14 ft 3. 18831.6 ft 4. 278.11 ft

This Monument is:

10/8 HAVE HW OUT Yesterday we did a survey Lab. See me to

make it up. You also finished warm ups and turned them in.

Today: Pick up Vector Note Sheet.

I have duty this pm until 2:50

Trig Quizzes should have been made up yesterday or this morning.

HAPPY BIRTHDAY RANDI W!

What is the difference between these tools?

We will be using a triangulation device like the bottom tool. Note that the center reference is 0º. This allows us to get the angle of elevation, not the zenith angle.

SURVEY LABHOW WOULD I FIGURE OUT HEIGHT?

This is same as reading on

triangulation tool. If using a

typical protractor, the angle

would represent the zenith.

This describes the

angle of elevation

45º

SURVEY LABHOW WOULD I FIGURE OUT HEIGHT?

angle of elevation = 45º

45º

This is same as elevation angle since 90-45-45 triangle

SOH CAH TOA

b

csin

opp

hyp

cos adj

hypa

c

tan opp

adjb

a

c a b2 2 2

TRIGONOMETRY REVIEW

We will be focusing on triangles

What is a right triangle? A triangle with a 90º angle What is a hypotenuse? Side of right triangle opposite the 90º angle

What is Pythagoreans Theorem? c2 = a2 + b2 where c is the hypotenuse.

Only applies to right triangles

EX A: GIVEN THE FOLLOWING TRIANGLE

a = 4.21u

b = 7.43 u

Angle C = 90.0°

What is the hypotenuse (c) ?

b

a

c

EX A: GIVEN THE FOLLOWING TRIANGLE

c2 = b2 + a2

c2 = 4.212 + 7.432

c = 8.54 u

How would you label the angles?

b

a

c

SAME TRIANGLE A

What is measure of smallest angle, θA?

θ is the Greek letter

theta and stands for angle

b

a

c

a = 4.21u

b = 7.43 u

c = 8.54 u

A

SOH CAH TOA

This is a good time to review SOH CAH TOA

What does sine, cosine, and tangent?

What does sine, cosine, and tangent represent?

The RATIO between given sides of a right triangle in reference to a specific angle.

SOH CAH TOA Triangle Demo

THE RATIOS….. Sine = opposite / hypotenuse Cosine = adjacent / hypotenuse

Tangent = opposite / adjacent

These only work for right triangles!

Show Table

Angle SinA CosA TanA

NAMING THE SIDES

A right angledtriangle

The angle weare interested in.

H

This is the longest side— the hypotenuse.

O

This side is oppositeour angle.

AThis side is adjacentto our angle.

NAMING THE SIDES

H = Hypotenuse

O = Opposite

A = Adjacent

O

NAMING THE SIDES

H = Hypotenuse

O = Opposite

A = Adjacent

H

O

A

O

H

AH

OA

HO

A H

O

A

EX B CONSIDER THIS TRIANGLE. WHAT IS THE SINE RATIO?

30°

4cm

8cmH =

O =

Opposite/Hypotenuse gives us the Sine Ratio.

sin 30° = 4cm/8cm = 0.5.

If the opposite side was 6 cm, what would the hypotenuse be?

If you enter Sin 30 in your calculator you should get 0.5. Try it! (sin button is in the trig menu)

EX C CONSIDER THIS TRIANGLE. WHAT IS THE ANGLE?

θ

5cm

12cmH =

O =

Name the sides in reference to the angle

Determine which trig function to use

Sin-1 (5/12) = 24.62º

To determine angle you use the inverse trig function for and enter the ratio of the corresponding sides.

Sin = O/H

Now go back to Example A and solve the angle using the inverse cosine function, then solve the angle using the inverse tan function

EXAMPLE A

a = 4.21u

b = 7.43 u

c = 8.54 uWhat is measure of smallest angle, θA?

Cos θA = adj/hyp

Cos-1 θA

(7.43/8.54)

θA = 29.54°

a

cb

A

EXAMPLE A

a = 4.21u

b = 7.43 u

c = 8.54 uWhat is measure of smallest angle, A?

Tan θ = opp/adj

Tan-1 θA

(4.21/7.43)

θ A = 29.54°

a

bc

A

HOW WOULD YOU DETERMINE THE LAST ANGLE B?

The sum of all angles in a triangle equals

180º 180º - 90º - 29.54º 60.46º

FOR RIGHT TRIANGLES If you know any two sides, you can determine the angle

If you know a side and an angle other than 90, you can determine a side

EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE?

22º

S

T

28 u


22º

S

T

28 u

Label Sides

opp

adj

What do you know? Hyp and angle

What function can you use to solve for opp? Sin = Opp/Hyp

Opp = Sin Hyp Opp = (Sin22º)(28u)

Opp = 10.49u


22º

S

T

28 u

How would you solve for side T?

10.49 u

adj

c2 = a2 + b2

I will call adjacent (T) side a and opposite (S) side b a2 = c2 - b2

a2 = (28u)2 – (10.49u)2

a = 25.96u


22º

S

T

28 u

How would you solve for the remaining angle *?

10.49 u

adj

Remember angles equal 180º

180º – 90º - 22º Angle * = 68º

*

Summary Putting it all together:

If you need to determine an angle :

Name sides in reference to angle of interest

Determine formula

You know opp and hypotenuse, want θ :

sin-1 = (Opp/Hyp) Use inverse function

sin-1 (5m/10m)=30º

5m opp

10m hyp

Θ ??

Summary Putting it all together:

If you need to determine a side:

Name sides in reference to known angle

Determine formula

You know angle and hypotenuse, want opposite:

Opp = (sinθ)(Hyp)

(sin30º)(10m) = 5m

?? opp

10m hyp

30º

Real World Applications

EX E THE SWIMMER

A swimmer attempts to swim due north to the pier 2.00 miles away but the current takes him at a bearing of 40°. After a while he notices he is due east of the pier. How far has he travelled?

Step 1. Draw a diagram.

pier

2.00

mile

s

40°?

EX E THE SWIMMER

?2 40°

Step 2. Identify the sides.

Here we have the Adjacent side and want to find the Hypotenuse. So we use the CAH triangle.

C H

A

Putting our finger on H shows that H = A/C= 2.00 ÷ (cos 40°)= = 2.61 miles

EX F FINDING AN ANGLE (1)

At Heathwick airport there is a forest just 500. m from the end of the runway. The trees can be as tall as 30. m. What is the minimum angle of climb if aircraft are to avoid the trees?


30.m

500.m?

EX F FINDING AN ANGLE (2)

30

500

Step 2. Identify the sides

Here we have the Adjacent and Opposite sides and want to find an angle. So, we use the TOA triangle.

Putting our finger on T shows that… tan = O/A

T AO

We can use the inverse tan to find the angle.= tan-1 (30m/500m) = 3.4°

EX G THE CHURCH STEEPLE

Eric decides to find the height of the steeple of his local church. He measures a distance of 50. m along the ground. The angle of elevation to the top of the steeple is 35°. How high is the steeple?


50.m 35°

?

THE CHURCH STEEPLE

?

50

35°

Step 2. Identify the sides.

Here we have the Adjacent side and want to find the Opposite. So, we use the TOA triangle.

Putting our finger on O shows that O = T × A

= (tan (35º) × 50m= 35.01 m

T AO

REMEMBER…

S H

OC H

A

T A

O

SOH-CAH-TOA

30°

? cm

8 cmH =

O =

S HO

SIN FINDING THE OPPOSITE

SOH-CAH-TOA? ?

Opp = Sin × Hyp

= (Sin 30°) × 8

= 4 cm

27°

? km

12.3 km

H =

A =

C HA

COS FINDING THE ADJACENT

SOH-CAH-TOA? ?

Adj = Cos × Hyp

= (Cos 27°) × 12.3

= 0.891 × 12.3

= 11.0 km

53°

? cm

T AO

TAN FINDING THE OPPOSITE

O =

A =

16 cm

SOH-CAH-TOA? ?

Opp = Tan × Adj

= (Tan 53°) × 16

= 1.327 × 16

= 21 cm

36°

87 m

? mH =

O =

S HO

SIN FINDING THE HYPOTENUSE

SOH-CAH-TOA??

Hyp = Opp Sin

= 87 (Sin 36°)

= 87 0.5878

= 150 m

0.80 cm

? cmH =

A =

C HA

COS FINDING THE HYPOTENUSE

60°SOH-CAH-TOA

? ?

Hyp = Adj Cos

= 0.80 (Cos 60.°)

= 0.80 0.50

= 1.6 cm

30°

3.1 cm T AO

TAN FINDING THE ADJACENT

O =

A = ? cm

SOH-CAH-TOA? ?

Adj = Opp Tan

= 3.1 (Tan 30.°)

= 3.1 0.5773

= 5.4 cm

WHAT HAPPENS WHEN YOU DON’T KNOW THE ANGLE?

We can find the usable number mentioned previously using the ratios.

The problem is we know need to convert it back into the original angle.

The Buttons on your calculator are…

Sin Cos Tan

The opposite of these are SHIFT then

Sin-1 Cos-1 Tan-1

3.0 km

7.0 kmH =

O =

S HO

SIN FINDING THE ANGLE

SOH-CAH-TOA? ??

Sin = Opp Hyp

Sin = 3.0 7.0

Sin = 0.4285

= Sin-1 (0.4285)

= 25°

12.1 cm

14.5cm

H =

A =

C HA

COS FINDING THE ANGLE

SOH-CAH-TOA? ??

Cos = Adj Hyp

Cos = 12.1 14.5

Cos = 0.834

= Cos-1 (0.834)

= 33.4 °

67.0 cm T AO

TAN FINDING THE ANGLE

O =

A = 187 cm

SOH-CAH-TOA? ??

Tan = Opp Adj

Tan = 67.0 187

Tan = 0.358

= Tan-1 (0.358)

= 19.7°

If two vectors are not at right angles to each other then we must use the Law of Cosines: C2 = A2 + B2 – 2AB cos

“” or Theta, is any unknown angle but in this case it is the angle between the two vectors

9/29 test wednesday pick up review & calculator have motion ws ii out warm up 2: look at the...

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