TrigonometryWorkbook

Name: ____________________________

Trigonometry Work Booklet

The work in the booklet is sequenced so that all students should be able to access tasks appropriate to their current learning.

Students must have a calculator

Skill 1 – Worksheet: Common Terms in Trigonometry

1. Maths that uses angles and side lengths in triangles to find missing sides and angles is known as …………………………..

2. What Greek letter is used to indicate the angle we are using when working with right angle triangles?

3. What are the different ways we can show the ratio between two lengths?

4. Using 3 and 6 create two different ratios

5. A trigonometric ratio is when we;

A. divide two sides B. compare two sides of a RAT for a given angleC. compare two sides of a RATD compare two sides of a RAT for any angle

6. In trigonometry the following symbols stand for;

O – H - A – 7. An abbreviation for a right angle triangle would be:

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Skill 2 Similar Triangles

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Skill 3 Exercises – Labelling Right Angle Triangles1. On the triangles below label the opposite (use O), adjacent (use A) and Hypotenuse (H). Remember to use the given angle as your point from which you identify the sides. The given angle is θ.

2.

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Skill 4 Exercise. Ratios in Right Angle Triangles: Identifying The Trigonometric ratios1. In the triangle below the given angle is dependent on the;A. ratio of the right angle and the other angleB. sides are all in ratioC. ratio of the two sides that make the angleD. ratio of the angles

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2. The Sin ratio is the ratio between the adjacent and hypotenuse sides. T or F

3. The Tan ratio is the ratio between the hypotenuse and adjacent sides. T or F

4. The Cos ratio = . T or F

5. Match the following

Sin Cos Tan

Skill 4A Exercises1. Find the ratio of the following trig ratios.

Skill 4B Exercises Using your inverse functions find the angles for sine, cosine and tangent of the following ratios

Sin Cos Tan0.34530.50.86601.0

øø

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Skill 5 Using Trigonometry to find Side Lengths

Skill 5A Exercise. Find the Side.Worked Example

1. For the following triangles identify which rule or ratio you will need to use

Remember: Label the diagram and write out the rule needed – see the above example

a. b.

c. d.

e. f.

25˚

25˚ 25˚

25˚

25˚

25˚

X

X

X

X

X

X16

16

16

16

16

16

35˚

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Opp The opp has an unknown value X and the Hyp is 12.

The Opp and Hyp are in ratio so I would use

Sin = to find the value of X

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Exercise 5B Finding the Top Line

EASY TO REMEMBER – When we have to find the missing number on top of the ratio TIMES the ANGLE by the NUMBER on TOP

Sin 35˚ = X = 15 x Sin 35˚

Cos 35˚ = X = 15 x Cos 35˚

Tan 35˚ = X = 15 x Tan 35˚

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Exercise 5C Finding the Bottom Line

Find the value of y correct to two decimal places.

EASY TO REMEMBER – When we have to find the missing number on the bottom of the ratio DIVIDE the NUMBER on TOP by the ANGLE

Sin 35˚ = X =

Cos 35˚ = X =

Tan 35˚ = X =

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Skill 5D Word Problems For the following express all answers correct to 2 decimal places

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Skill 6 Using Trigonometry to find AnglesWorked Example

a. Label the sides (O, A, H ) using the given angleb. Identify which of the labelled sides have a measurement or pronumeral c. Write down the required rule

NOTE: The same process applies for Cos and Tan triangles

1. For the following triangles identify which rule or ratio you will need to use

Remember: Label the diagram and write out the rule needed – see the above example

a) b)

c) d)

e) f)

12

X˚ X˚12 77

OppHyp The opp has a known value X

and the Hyp is 12.

The Opp and Hyp are in ratio so I would use

Sin = to find the value of X˚

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28 56

27

25

25˚

X X

X

X˚

X

1616

16

16

16

16

X˚

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Skill 7 Exercises. Angles of Depression and Elevation1. A rescue helicopter spots a missing surfer drifting out to sea on his damaged board. The helicopter descends vertically to a height of 19 m above sea level and drops down an emergency rope, which the surfer grips. Due to the wind the rope swings at an angle of 27° to the vertical, as shown in the diagram. What is the length of the rope?

2.

3.

4. The angle of elevation of the top of a tree from a point on the ground, which is 60 m from the tree, is 35°.a) Draw a labelled diagram, which represents the situation.b) Find the height of the tree.

5. From a rescue helicopter 1800 m above the ocean, the angles of depression of two shipwreck survivors are 40° and 60° respectively.a) Draw a labelled diagram, which represents the situation.b) Calculate how far apart the two survivors are.

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Skill 8 Exercises. Bearings

1.

2.

3. A ship which was to travel due north, veered off course, and travelled N 80°E (or 080°T) for a distance of 280 km as shown in the figure at right.a. How far east had the ship travelled?

b. How far north had the ship travelled?

4. A hiker walks 2.4 km east, then walks S75°W until she is due south of her starting position.a. Draw a diagram og the scenario.b. How far is she from her starting point?c. How far did she walk on the bearing of S75°W before she was due south of her starting position?

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Skill 9 Applications of Trigonometry

1. What angle does a 3.8 m ladder make with the ground, if it reaches 2.1 m up the wall? How far is the foot of the ladder from the wall?

2. While fishing at Lake Eildon, Michael anchors his boat with a 60 m long cable which is inclined at 65° to the vertical as shown in the figure. What is the depth of the water?

3. Find the values of the pronumerals in each of the following triangles, using appropriate trigonometric ratios. Give side lengths correct to 2 decimal places and angles to the nearest degree.

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4. From a point on top of a cliff, two boats are observed. If the angles of depression are 58° and 32° and the cliff is 46 m above sea level, how far apart are the boats?

5. A 2.05 m tall man, standing in front of a street light 3.08 m high, casts a 1.5 m shadow.

a. Draw a diagram of this situationb. What is the angle of elevation from the ground to the source of light?c. How far is the man from the bottom of the light pole?

6. Joseph is asked to obtain an estimate of the height of his house using any mathematical technique. He decides to use an inclinometer and basic trigonometry. Using the inclinometer, Joseph determines the angle of elevation, from his eye level to the top of his house to be 42°. The point from which Joseph measures the angle of elevation is 15 m away from his house and the distance from Joseph’s eyes to the ground is 1.76 m.

a. Fill in the given information on the diagram provided (substitute values for the pronumerals).b. Determine the height of Joseph’s house.

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Answers:Skill11. Trigonometry, 2. Θ, 3. Ratio and Fraction, 4. 3:6, 6:3, 5. B, 6. Opposite, Hypotenuse, Adjacent, 7. RAT

Skill 21a. SAS, b. AAA, c.RHS, d. SSS2a. 14cm, b.8m c. 7.5cm, d. 20mm3a B, b. C, 4a.C, b.C

Skill 3

Skill 41. C, 2.F, 3.F, 4.F, 5. Sin=O/H, Cos=A/H, Tan = O/A

Skill 4A1a. 0.7314, b. 0.9613, c. 1.234, d. 0.9962, e. 0.5460, f. 0.9063, g. 0.5773, h. 0.9903Skill 4B

Sin Cos Tan0.3453 20.2 69.80 19.050.5 30 60 26.570.8660 60 30 40.891.0 90 0 45

Skill 51a. Sin, b. Sin, c. Cos, d. Tan, e. Tan, f. Sin

Skill 5B1a. 26.63, b. 7.51, c. 18.87, d. 33.27, e. 4.04, f. 0.92. 2a. 313.49cm, b. 137.73mm, c. 13.62m, d. 229.40km, e. 89.71cm, f. 150.13m.3a. C, b. D. 4a. C, b. D

Skill 5Da. 2.40m, b. 1.39m, c.7.60m, d. 10.72m

Skill 61a. Sin, b. Cos, c. Cos, d. Tan, e. Tan, f. Tan

1a. 15°, b. 39°, c. 35°, d. 62°, e. 10°, f. 41°. 2. 54°, 3a. B, b. C, d. D

Skill 71. 21.32m, 2. 122.06m, 3. 16, 4. 42.01m, 5. 1105.93m,

Skill 81. 34.64km, 2. B, 3a. 275.75km E, b. 48.62km N, 4a. ,b. 0.64 km, c. 2.48km,

Skill 91. 33.56, 2. 25.36m, 3a. Θ=36.87°, a=15, β=56.44, 4°. 3b. x=41.57, y=121.54, 3c. y=56.31°, x=29.98°, z=30.02, 4. Boats are 28.74m and 73.62m from cliff, dist between boats in 44.88m. 5b. 53.81°, c.2.25m. 6. X=13.51m + 1.76m = 15.27m

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