FOM2 - UNIT 4 – NOTE PACKETFOM2 Unit 4 Notes: DAY 1 – SOH CAH TOA Warm Up

Find the length of the missing side in the following examples using the Pythagorean Theorem. Round answers to the nearest hundredth (2 decimal places), if necessary.

Pythagorean Theorem : a2+b2=c2

1. 2. 3.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii

1

3cm

6cm X

22cm

14cm

X

4cm

7cmX

Day 1 Notes: Trigonometric Ratios Lesson Handout

The word trigonometry comes from 2 Greek words, trigon, meaning triangle, and metron, meaning measure. The study of trigonometry involves triangle measurement.

We will be studying basic _________________ ____________________ _______________________

Identifying Sides

Hypotenuse side ___________

Opposite of angle A _________ Opposite of angle B________

Adjacent to Angle A _________ Adjacent to Angle B _______

The trig ratios we will be studying are ________________, __________________, and ______________________

The ratios for each function:

Sin =___________________ Cos = ____________________ Tan = ____________________

Setting up Ratios

Sin A = __________ Sin B = _________

Cos A = __________ Cos B = _________

Tan A = __________ Tan B = _________

Sin X = __________ Sin Y = _________

Cos X = __________ Cos Y = _________

Tan X = __________ Tan Y = _________

2

A

b

aC B

c

A

3

4C B

5

13

Z Y5

12

X

Using a Calculator (Remember: Calculator must be in _________________ mode)

Sin 39 = _________ Cos 58 = _________ Tan 85 = _________ Sin 30 = _________

Steps for solving for a missing side:

1. Find given ______________________ (Non-right angle)

2. Label sides __________________________, _________________________, ___________________________.

3. Write ratio using SOH CAH TOA

a. SOH ______________________________

b. CAH ______________________________

c. TOA ______________________________

4. Find Sine, Cosine, or Tangent in calculator.5. Cross multiply.6. Solve for missing variable.

Solving for a Side

1. 2.

3. 4.

3

38°

10

X

50°

A4570°z

34

85°12

Y

Day 1 Trigonometric Ratios Practice

1. Find sin A.

1. Mark ∡ A. 2. Label the sides in relation to ∡ A (opp, adj, hyp)3. Circle the sides that are needed to find sin (opp, hyp)4. Write the sin ratio for ∡ A5. Change the ratio to decimal form.

Sin A = _________

2. Find BC

1. Mark angle with the given value.2. Label the sides in relation to the given angle (opp, adj, hyp)3. Circle the known relationships. (adj, hyp)4. Decide which trig function uses these two relationships.5. Write the trig equation used to solve this problem. 6. Solve the equation.

BC = _________

3. Find the measure of each side indicated. Round your answer to the nearest tenth. a. b.

4. Suppose you’re flying a kite, and it gets caught at the top of the tree. You’ve let out all 100 feet of string for the kite, and the angle that the string makes with the ground is 75 degrees. Instead of worrying about how to get your kite back, you wonder. “How tall is that tree?”

FOM2 Unit 4 Notes: DAY 2 – SOH CAH TOA (Inverse) Warm Up

4

A

B

C

5

12

13

AC

x17

B50°

h

75°

100 ft

The tailgate of a moving van is 3.5 feet above the ground. A loading ramp is attached to the rear of the van. The angle that the ramp makes with the ground is 10°. Find the length of the loading ramp to the nearest tenth of a foot.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii

Day 2 Inverse Trig Ratios

What is an inverse?

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What do we use the inverse for?

Our goal when solving an equation is to isolate our variable (this means get it by itself)

Inverse Trig Functions

sin θ→ sin−1θ cosθ→cos−1θ tanθ→ tan−1θ

Ex) Find x for sin(x) = 0.5431.

What is the x representing? When do we use inverse trig?

Using Trigonometry to Find Missing Angles of Right Triangles

Calculator Instructions: “2nd” “Tan or Cos or Sin”

Ex. Use the inverse functions on your calculator to evaluate the following. Round your answers to the nearest hundredth of a degree.

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a) cos(x) = 0.5431 x ≈ ____________

b) tan(x) = 0.5431 x ≈ ____________

c) sin( θ ) = 0.8426 ≈ θ ____________

d) tan(B) = 3.5 B ≈ ____________

e) sin( θ ) = 37 ≈ θ ____________

f) cos(A) = 413 A ≈ ____________

Using Trigonometry to Find Missing Angles of Right Triangles

(Note: Figures in this section may not be drawn to scale.)

1. Let us examine the following triangle, and learn how to use Trigonometry to find x.

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Steps:a) Since x is the angle that we want to find, we will let this angle be our reference angle. Using x as

your reference, find what sides are given.

b) Write Ratio.

b) Take inverse.

c) What if we now wanted to find the measure of the other acute angle of the triangle? Let us name it y. We certainly could use Trigonometry again, but can you think of a faster way to find the value of y?

Examples: Find the indicated angles in each of the triangles below. Round your final answers to the nearest hundredth. (Figures may not be drawn to scale.)

a) b)

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c) d)

e)

Ex. Find ALL the missing side and angle measures of the triangles below. Round your answers to the nearest hundredth.

a) b)

FOM2 Unit 4 Notes: DAY 3 – SOH CAH TOA (Practice)

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10

FOM2 Unit 4 Notes: DAY 4- Applications Warm-up

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1. Find each angle measure to the nearest degree

a. tan A = 2.0503 b. cos Z = 0.1219

c. sin U = 0.8746

2. Find the measure of the indicated angle to the nearest degree.

a. b.

c. d.

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Day 4 Notes

Trig Applications are solve by using ________________________________.

Steps:

1) Draw a picture.2) Label the picture accurately3) Set up proper Trig Ratio4) Solve

Examples:

1) To see the top of a building 1000 feet away, you look up 28 from the horizontal. What is the height of the building?

2) A guy wire is anchored 12 feet from the base of a pole. The wire makes a 62 angle with the ground. How long is the wire?

Day 4 Applications –Practice

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1. Two legs of a right triangle are 16 and 48. Find the measure hypotenuse.

2. One leg of a right triangle is 14 while the hypotenuse is 38. Find the measure of all the angles.

3. The bottom of 24-foot ladder is 6 feet from the building that the ladder is leaning against. In order for the ladder to be set-up safely the angle the ladder makes with the ground cannot exceed 75°. Is the ladder set up safely? How do you know?

4. A jet airplane out of Denver, Colorado needs to clear a 1,500 ft mountain 1 mile (5,280 feet) after it takes off. If the plane makes a steady climb after takeoff, what angle does the plane need to take to clear the mountain?

5. The Washington Monument is 555 feet tall. An observer is 300 feet from the base of the monument. If the observer is lying on the ground looking to the top of the monument, find the angle made between the observer’s line of sight and the ground.

6. Jenna goes on an exciting airplane ride. She takes off at a 25 angle and continues flying in a perfectly straight path until she is directly over her house, as shown. She notices that her altitude when directly over her house is 3,200 feet. What distance has she flown?

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7. Carl decides to use what he has learned about trigonometry to help him find the height of his favorite tree. At a certain time of day, he measures the tree’s shadow with a tape measure and finds that it is 31 feet long. Then he measures the angle of elevation to the sun using a clinometer and finds that the sun’s rays are striking at a 62 angle with the ground. (The angle that the sun’s rays make as they strike an object determines the length of the object’s shadow.) Use the information and what you know about trigonometry to calculate the height of the tree.

8. Erica is standing on one side of a canyon, and her friend Sasha is standing directly across the canyon from her on the other side. They want to know how wide the canyon is. Erica marks her spot and then walks 10 yards along the canyon edge and looks back at Sasha. The angle of her line of sight to Sasha and the path she just walked is 72. Draw a sketch that illustrates this situation. What is the approximate width of the canyon?

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10)An evergreen tree is supported by a wire extending from 1.5 feet below the top of the tree to a stake in the ground that is 15 feet from the base of the tree. The wire forms a 58 angle with the ground. How tall is the tree?

11)To the nearest tenth of a foot, how tall is a building 100 feet away (d = 100) if the top of the building is sighted at a 20° angle (n = 20)?

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9.

12)If an object is dropped from the top of the leaning tower of Pisa, it will land about 13 feet from the base of the tower. The tower leans at an angle of approximately 86°. How far did the object drop?

13)A ramp was built by the loading dock of a building. The height of the loading dock platform is 7 feet. Determine the length of the ramp if it makes a 38 angle with the ground. (Draw a picture!)

14)A jet airplane begins a steady climb of 15˚ and flies for two ground miles. What was its change in altitude?

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13 ft.

d

86°

FOM2 Unit 4 Notes: DAY 5- Applications Practice

Warm-upUse trigonometric ratios to find the value of the variables. Round to the nearest hundredth.

A) B)

C) D)

FOM2 Unit 4 Notes: DAY 6- Angles of Elevation & Depression 18

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x

5

1) Jenna goes on another exciting airplane ride. She takes off at a 35° angle and continues flying in a perfectly straight path for five miles. She discovers that she is directly over her house.

a) How far is her house from the airport?

b) What is her altitude?

Day 6 Notes

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35

5 mi

Angle of___________________________________: the angle formed by a __________________________________________ and the line of sight to an object above the horizontal line.

Angle of ___________________________________: the angle formed by a __________________________________________ and the line of sight to an object below the horizontal line.

Ex 1) From a point 80 m. from the base of a tower, the angle of elevation to the top of the tower is 28o. How tall is the tower?

Ex 2) A ladder that is 20 ft. long is leaning against the side of a building. If the angle formed between the ladder and the ground is 75o. How far is the bottom of the ladder from the base of the building?

Ex 3) When the sun is 62o above the horizon, a building casts a shadow 18 m. long. How tall is the building?

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Ex 4) The angle of depression from the top of a tower to a boulder on the ground is 38o. If the tower is 25 m. high, how far from the base of the tower is the boulder?

Ex 5) An observer at the top of the building sees a car on the road below. The angle of depression to the car is 28o. If the car is 50 m. from the building when it is seen, how tall is the building?

Day 6 Angles of Elevation and Depression Practice

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1 At a certain time of day the angle of elevation of the sun is 44°. Find the length of the shadow cast by a building 30 meters high.

2 The top of a lighthouse is 120 meters above sea level. The angle of depression from the top of the lighthouse to the ship is 23°. How far is the ship from the foot of the lighthouse?

3 A lighthouse is 100 feet tall. The angle of depression from the top of the lighthouse to one boat is 24°. The angle of depression to another boat is 31°. How far apart are the boats?

4. At a point on the ground 100 ft. from the foot of a flagpole, the angle of elevation of the top of the pole contains a 31 degree angle. Find the height of the flagpole to the nearest foot.

5. From the top of a lighthouse 190 ft. high, the angle of depression of a boat out at sea is 34 degrees. Find to the nearest foot, the distance from the boat to the foot of the lighthouse.

6. Find to the nearest degree the measure of the angle of elevation of the sun if a post 5 ft. high casts a shadow 10 ft. long.

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