Unit 4 Trigonometry

Unit 4 Trigonometry Lesson Date In-Class

1 10/2

Fri. Right Triangle Trig

2 10/5

Mon. Angle of Elevation and Depression

3 10/6

Tue. Law of Sines

10/7

Wed. No Virtual Class

4 10/8

Thurs. Law of Cosines

5 10/9

Fri. Application of Law of Sines and Cosines

6 10/12

Mon. Graphs of Sine and Cosine

7 10/13

Tue. Sinusoidal Transformations

10/14

Wed. No Virtual Class

8 10/15

Thurs. Review

9 Fri.

10/16 Test

Day 1: Things to recall from Geometry about right triangles:

• The angles of a triangle add up to _________ degrees.

• The side across from the right angle is called the ___________. This is always the longest side.

• = theta, the measure of an angle (could be degrees or radians but we will use degrees for right now).

• Each angle is represented with a capital letter and its corresponding side is represented by the lower case letter of that.

Pythagorean Theorem

a. Pythagorean Theorem is used to find missing sides in a triangle.

b. “a” and “b” represent the

c. “c” represents the

d. Examples: Find the missing sides using Pythagorean Theorem

i. 2.

3. 4.

Labeling Triangles

In a triangle, depending on where theta is, depends on where the side that is opposite or adjacent is. Opposite – straight across from (opposite of it), and adjacent – right next to. Ex:

7

10

SOHCAHTOA SOHCAHTOA is used to help find missing sides and angles in a right triangle when

Pythagorean Theorem does not work!

S (cosine) O (cosine) H (hypotenuse) →

C (cosine) A (cosine) H (hypotenuse) →

T (cosine) O (cosine) A (hypotenuse) →

CSC=----------- SEC= ------------ COT=-----------

Given the following, find the six trig ratios. (This means, don’t find angles, just set up the sides based on the ratios under SOH-CAH-TOA).

1.

8

2. Find all 6 trig ratios if sin𝜃= 5/13

Finding Missing Sides of Triangles

Setting up Trigonometry Ratios and Solving for Sides

i. (NOT the right angle)

ii. (Opposite, Adjacent, Hypotenuse)

iii. :

✓ if we have the opposite and hypotenuse

✓ if we have the adjacent and the hypotenuse

✓ if we have the opposite and the adjacent

iv. Set up the proportion and solve for x!

Example:

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Kuta Software - Infinite Geometry Name___________________________________

Period____Date________________Solving Right Triangles

Find the missing side. Round to the nearest tenth.

1)

6

x

72°

2)

x

6

73°

3)

x

12

24°

4)

x

12

37°

5)

14

x

49°

6)

14x

51°

7)

16x

63°

8)

16x

15°

-1-

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9)

29 x

55°

10) x

21

68°

11) x

2919°

12)

x22

21°

13)

x

29

33°

14)

35

x

45°

15)

28x47°

16)

34

x

59°

Critical thinking question:

17) Write a new problem that is similar to the

others on this worksheet. Solve the

question you wrote.

-2-

Setting up Trigonometry Ratios and Solving for Angles

i. Example: Find the measure of angle A

Practice:

I. Find the missing side or angle. 1.

X

30° 8 8

2. 40°

x

11 3.

13

II. If it says to SOLVE the triangle, you want all 3 sides and all 3 angles.

1.

B

C

A

48°°

12

8

Day 2: Angle of Elevation/Depression Application Problems

Angle of Elevation Angle of Depression

1. A tree casts a 5m shadow. Find the height of the tree if the angle of elevation of the sun is 32.3º.

Sketch: Work: Answer:

2. A ladder 10.4 m long leans against a building that is 1.5 meters away. What is the angle formed by the

ladder and the building?


3. A ladder 8.6 m long makes an angle of 68º with the ground as it leans against a building. How far is the foot

of the ladder from the foot of the building?


4. The angle of depression from the top of a cliff 800 meters high to the base of a log cabin is 37º. How far is

the cabin from the foot of the cliff?


6. From a point on the ground 500 ft from the base of a building, it is observed that the angle of elevation to

the top of the building is 24º and the angle of elevation to the top of a flagpole atop the building is 27º. Find the

height of the building and the length of the flagpole.


5. Mrs. Roberts stands 25ft from the flagpole. She looks and the angle of elevation to the top of the flagpole

is 45 degrees. Find the height of the flagpole.


Geometry Worksheet Name__________________________________

8.5 (Angles of Elevation & Depression) Date___________________Period__________

Draw a picture, write a trig ratio equation, rewrite the equation so that it is calculator ready and then

solve each problem. Round measures of segments to the nearest tenth and measures of angles to the

nearest degree.

________1. A 20-foot ladder leans against a

wall so that the base of the ladder is 8 feet from

the base of the building. What is the ladder’s

angle of elevation?

________2. A 50-meter vertical tower is

braced with a cable secured at the top of the

tower and tied 30 meters from the base. What

is the angle of depression from the top of the

tower to the point on the ground where the

cable is tied?

________3. At a point on the ground 50 feet

from the foot of a tree, the angle of elevation to

the top of the tree is 53. Find the height of the

tree.

________4. From the top of a lighthouse 210

feet high, the angle of depression of a boat is

27. Find the distance from the boat to the foot

of the lighthouse. The lighthouse was built at

sea level.

________5. Richard is flying a kite. The kite

string has an angle of elevation of 57. If

Richard is standing 100 feet from the point on

the ground directly below the kite, find the

length of the kite string.

________6. An airplane rises vertically 1000

feet over a horizontal distance of 5280 feet.

What is the angle of elevation of the airplane’s

path?

________7. A person at one end of a 230-foot

bridge spots the river’s edge directly below the

opposite end of the bridge and finds the angle

of depression to be 57. How far below the

bridge is the river?

________8. The angle of elevation from a car

to a tower is 32. The tower is 150 ft. tall.

How far is the car from the tower?

230

________9. A radio tower 200 ft. high casts a

shadow 75 ft. long. What is the angle of

elevation of the sun?

________10. An escalator from the ground

floor to the second floor of a department store

is 110 ft long and rises 32 ft. vertically. What

is the escalator’s angle of elevation?

________11. A rescue team 1000 ft. away

from the base of a vertical cliff measures the

angle of elevation to the top of the cliff to be

70. A climber is stranded on a ledge. The

angle of elevation from the rescue team to the

ledge is 55. How far is the stranded climber

from the top of the cliff? (Hint: Find y and w

using trig ratios. Then subtract w from y to

find x)

________12. A ladder on a fire truck has its

base 8 ft. above the ground. The maximum

length of the ladder is 100 ft. If the ladder’s

greatest angle of elevation possible is 70,

what is the highest above the ground that it can

reach?

________13. A person in an apartment

building sights the top and bottom of an office

building 500 ft. away. The angle of elevation

for the top of the office building is 23 and the

angle of depression for the base of the building

is 50. How tall is the office building?

________14. Electronic instruments on a

treasure-hunting ship detect a large object on

the sea floor. The angle of depression is 29,

and the instruments indicate that the direct-line

distance between the ship and the object is

about 1400 ft. About how far below the

surface of the water is the object, and how far

must the ship travel to be directly over it?

110

32

1000

x

w y

8

100

500

1400

17

Law of Sines

Law of sines suggests that in ANY triangle (not just right triangles) the lengths of the sides are

proportional to the sines of the corresponding opposite angles.

Formula:

All are equal! It doesn’t matter which ones you use!

Use the law of sines if you are given a triangle with:

Side-Angle-Angle (SAA)

Side-Side-Angle (SSA)- Ambiguous case (MONDAY)

Examples

1. SAA (Side-angle-angle)

2. SAA

3. SAA

4. A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los

Angeles, 340 miles apart. At an instant when the satellite is between these two stations, its angle of

elevation is simultaneously observed to be 600 at Phoenix and 750 at Los Angeles. How far is the

satellite from Los Angeles

18

Day 3: Law of Sines Practice

23

Law of Cosines

Law of Cosines is used when you are given the following information:

SSS- Know all three sides and no angles

SAS- Know 2 sides and the angle between them

Example 1: In ΔABC, m∠A=39, AC=21 and AB=42. Find side a to the nearest integer.

Example 2: In the triangle below, find the measure of angle x.

To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245

yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake.

Approximately how long is the lake?

25

26

27

Law of Sines/Cosines Word Problems 1. A post is supported by two wires (one on each side going in opposite directions) creating an angle

of 80° between the wires. The ends of the wires are 12m apart on the ground with one wire forming an angle of 40° with the ground. Find the lengths of the wires.

2. Two ships are sailing from Halifax. The Nina is sailing due east and the Pinta is sailing 43° south of east. After an hour, the Nina has travelled 115km and the Pinta has travelled 98km. How far apart are the two ships?

3. 3 friends are camping in the woods, Bert, Ernie and Elmo. They each have their own tent and the tents are set up in a Triangle. Bert and Ernie are 10m apart. The angle formed at Bert is 30°. The angle formed at Elmo is 105°. How far apart are Ernie and Elmo?

4. Two scuba divers are 20m apart below the surface of the water. They both spot a shark that is below them. The angle of depression from diver 1 to the shark is 47° and the angle of depression from diver 2 to the shark is 40°. How far are each of the divers from the shark?

5. To estimate the length of a lake, Caleb starts at one end of the lake and walks 95m. He then turns and walks on a new path, which is 120° to the direction he was first walking in, and walks 87m more until he arrives at the other end of the lake. Approximately how long is the lake?

6. Two observers are standing on shore ½ mile apart at points F and G and measure the angle to a sailboat at a point H at the same time. Angle F is 63° and angle G is 56°. Find the distance from each observer to the sailboat.

7. Jack and Jill both start at point A. They each walk in a straight line at an angle of 105° to each other. After 45 minutes Jack has walked 4.5km and Jill has walked 6km. How far apart are they?

8. Points A and B are on opposite sides of the Grand Canyon. Point C is 200 yards from A. Angle B measures 87° and angle C measures 67°. What is the distance between A and B?

9. A 4m flag pole is not standing up straight. There is a wire attached to the top of the pole and anchored in the ground. The wire is 4.17m long. The wire makes a 68° angle with the ground. What angle does the flag pole make with the wire?

Graphing Trigonometric Functions Vocabulary

1. Periodic:

2. Sinusoidal:

3. Fundamental Period:

4. Amplitude:

5. Vertical Shift:

6. Phase Shift:

7. Domain (for sine & cosine):

8. Range (for sine & cosine):

9. Maximum (for sine & cosine):

10. Minimum (for sine & cosine):

Complete the table of values and then graph the functions.

x = y = sin

0

30

45

60

90

120

135

150

180

210

225

240

270

300

315

330

360

What’s important about a sine graph?

sine parent function:

Complete the table of values and then graph the functions.

x = y = cos

0

30

45

60

90

120

135

150

180

210

225

240

270

300

315

330

360

What’s important about a cosine graph?

cosine parent function:

Homework:Find the amplitude, domain, range, yintercept, xintercepts, maximums, and minimums of each function. Then, write an equation of the function.

1)

2)

Find the amplitude of the function and sketch its graph. Then, find the domain, range, yintercept, xintercepts, maximums, and minimums.

3)

4) Observe the function below:a) What is the range of the function?

b) What is the amplitude of the function?

c) What might be the equation of the function?

5) Observe the function below:

a) Write an expression that represents all of the graph's xintercepts.

b) Write an expression that represents all of the graph's maximums.

Practice

22

Graphs of Sine and Cosine From your discovery activity yesterday, you should have discovered that sine and cosine values repeat themselves. Thus, the sine and cosine functions are ___________________. A function is ____________________ if there is a positive number p such that f(t + p) = f(t) for every t. The least such positive number is called the __________. Points you should know from the unit circle

x Angle in Radians

0 π/2 π 3π/2 2π

y sin x

x Angle in Radians

0 π/2 π 3π/2 2π

y cos x

23

The sine and cosine curves

𝑦 = 𝐴 sin(𝐵(𝑥 − 𝐶)) + 𝐷 𝑎𝑛𝑑 𝑦 = 𝐴 cos(𝐵(𝑥 − 𝐶)) + 𝐷 The value A affects the amplitude. The amplitude (half the distance from the max to the min) will be |A| because distance is always _____________________. Increasing or decreasing an A value with vertically ______________ or ___________________ a graph.

The change in amplitude changes the “height” but not the width. This graph still reaches

from 0 to 2π.

The B value is the number of cycles it completes in an interval of ________________ or ________.

The B value affects the period. The period of sine and cosine is |2𝜋

𝑏|. When 0<B<1 the

period of the function is _______________ than 2π and the graph will have a _____________________

_____________________. When B>1, the period is _________________ than 2π and the graph will

have a horizontal ______________________.

27

Just like any other function, adding a constant on the end of the function will shift the trig graph _________________________ (up if the constant is _________________, down if the constant is ___________________).

To determine the midline of a graph you can add the max and min and divide by two.

Just like any other function, adding a constant on into the function will shift the trig graph _________________________ (left if the constant is _________________, right if the constant is ___________________). This is called a ______________________.

Note: You may have to factor out B in order to determine the phase shift.

Graphing Trig Functions

𝒚 = 𝑨 𝐬𝐢𝐧(𝑩(𝒙 − 𝑪)) + 𝑫 𝒚 = 𝑨 𝐜𝐨𝐬(𝑩(𝒙 − 𝑪)) + 𝑫

|A| = amplitude B = Horizontal Stretch, use to find

period |2𝜋

𝐵|

C= Phase Shift (Horizontal Shift) D= Vertical Shift (or

“displacement”)

Translations of Sinusoidal FunctionsPractice:

1) 2)

3) 4)

5)

6)

7)

8)

9)

10)

11)

Unit 4 Trigonometry

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