6.2 what is trigonometry? the hypotenuse is opposite the right angle. adjacent hypotenuse! opposite...

176 Maths Quest 9

MEASUREMENT AND GEOMETRY

6.2 What is trigonometry? The word trigonometry is derived from the Greek

words trigonon (triangle) and metron (measurement).Thus, it literally means ‘to measure a triangle’.

Trigonometry deals with the relationship between the sides and the angles of a triangle.

Modern day uses of trigonometry include surveying land, architecture, measuring distances and determining heights of inaccessible objects.

In this chapter relationships between the sides and angles of a right-angled triangle will be explored.

Naming the sides of a right-angled triangle The longest side of a right-angled triangle

(the side opposite the right angle) is called the hypotenuse.

In order to name the remaining two sides another angle, called the ‘reference angle’, must be added to the diagram.

The side that is across from the reference angle, !, is called the opposite side, and the remaining side (the side next to the reference angle) is called the adjacent side.

Note: If there is no reference angle marked, only the hypotenuse can be named.

Label the sides of the right-angled triangle shown using the words hypotenuse, adjacent and opposite.

THINK WRITE/DRAW

1 The hypotenuse is opposite the right angle.

Adjacent

Hyp

oten

use

θ

Opp

osite

2 Label the side next to angle θ as ‘adjacent’ and the side opposite angle θ as ‘opposite’.

HypotenuseHypotenuseOpposite

Adjacentθ

WORKED EXAMPLE 1

θ

178 Maths Quest 9


Trigonometric ratios Trigonometry is based upon the ratios between pairs of side lengths, and each one is

given a special name as follows.In any right-angled triangle:

sine (θ) = opposite

hypotenuse

cosine (θ) = adjacent

hypotenuse

tangent (θ) = opposite

adjacent

These rules are abbreviated to:

sin(θ) = OH

, cos(θ) = AH

and tan(θ) = OA

.

The following mnemonic can be used to help remember the trigonometric ratios. SOH − CAH − TOA

For this triangle, write the equations for the sine, cosine and tangent ratios of the given angle.

513

12θ

THINK WRITE/DRAW

1 Label the sides of the triangle.

5Opposite

Hypotenuse

Adjacent

13

12θ

2 Write the trigonometric ratios. sin(θ) = OH

, cos(θ) = AH

, tan(θ) = OA

3 Substitute the values of A, O and H into each formula.

sin(θ) = 513

, cos(θ) = 1213

, tan(θ) = 512

int-0744

Adjacent

Opp

osite

Hypot

enus

e

θ

H

H

AA

O

O

SOH

CAHTOA

θ

θθ

WORKED EXAMPLE 2

Trigonometry 181


5 WE2 For each of the following triangles, write the expressions for ratios of each of the given angles:i sine ii cosine iii tangent.

a 6

10 8

θ b

i h

gα

c

0.91.5

1.2

β

d 7

24

25

γ e

a b

cβ

f

v u

tγ

6 WE3 Write the trigonometric ratio that relates the two given sides and the reference angle in each of the following triangles.

a

1215

θ

b 25

30

θ

c 5

4

θ

d p

2.7

θ

e

t17

35°

f 14.3

α17.5

g

x

7

15°

h

20 31

θ i

9.8

3.1α

UNDERSTANDING

7 MC a What is the correct trigonometric ratio for the triangle shown at right?

A tan(γ) = ac B sin(γ) = c

a C cos(γ) = cb

D sin(γ) = cb

b a

c

γ

Trigonometry 183


6.3 Calculating unknown side lengthsValues of trigonometric ratios The values of trigonometric ratios can be found using a calculator. Each calculator has several modes. For the following calculations, your calculator must

be in degree mode.

Evaluate each of the following, giving answers correct to 4 decimal places.a sin(53°) b cos(31°) c tan(79°)THINK WRITE

a 1 Set the calculator to degree mode. Write the first 5 decimal places.

a sin(53°) = 0.798 63

2 Round correct to 4 decimal places. ≈ 0.7986

b 1 Write the first 5 decimal places. b cos(31°) = 0.857 16


c 1 Write the first 5 decimal places. c tan(79°) = 5.144 55


Finding side lengthsIf a reference angle and any side length of a right-angled triangle are known, it is possible to find the other sides using trigonometry.

Use the appropriate trigonometric ratio to find the length of the unknown side in the triangle shown. Give your answer correct to 2 decimal places.THINK WRITE/DRAW

1 Label the given sides.

x58°

16.2 m

Opposite

Adjacent

2 These sides are used in TOA. Write the ratio. tan(θ) = OA

3 Substitute the values of θ, O and A into the tangent ratio.

tan(58°) = x16.2

4 Solve the equation for x. 16.2 × tan(58°) = x x = 16.2 tan(58°)

5 Calculate the value of x to 3 decimal places, then round the answer to 2 decimal places.

x = 25.925 x ≈ 25.93 m

CASIOTIWORKED EXAMPLE 4


x58°

16.2 m

184 Maths Quest 9


Find the length of the side marked m in the triangle at right. Give your answer correct to 2 decimal places.

THINK WRITE/DRAW

1 Label the given sides.17.4 cmAdjacent

Hypotenusem

22°

2 These sides are used in CAH. Write the ratio. cos(θ) = AH

3 Substitute the values of θ, A and H into the cosine ratio.

cos(22°) = 17.4m

4 Solve for m: Multiply both sides by m. Divide both sides by cos(22°).

m cos(22°) = 17.4

m = 17.4cos (22°)

5 Calculate the value of m to 3 decimal places, then round the answer to 2 decimal places.

m = 18.766 m ≈ 18.77 cm

Benjamin set out on a bushwalking expedition. Using a compass, he set off on a course N 70°E (or 070°T) and travelled a distance of 5 km from his base camp.

N

E5 km

Base camp

70°

a How far east has he travelled?b How far north has he travelled from the base camp?Give answers correct to 2 decimal places.

THINK WRITE/DRAW

a 1 Label the easterly distance x.Label the northerly distance y.Label the sides of the triangle: Hypotenuse, Opposite, Adjacent.

a

5 kmHypotenuse

Opposite

Adjacent

x

y

70°


17.4 cm

m22°

WORKED EXAMPLE 7

Trigonometry 185


2 To calculate the value of x, use the sides of the triangle: x = O, 5 = H.These are used in SOH. Write the ratio.

sin(θ) = OH

3 Substitute the values of the angle and the pronumerals into the sine ratio.

sin(70°) = x5

4 Make x the subject of the equation. x = 5 sin(70°)5 Evaluate x to 3 decimal places, using a

calculator. = 4.698

6 Round to 2 decimal places. ≈ 4.70 km7 Answer the question in sentence form. Benjamin has travelled 4.70 km

east of the base camp.

b 1 To calculate the value of y, use the sides: y = A, 5 = H.These are used in CAH. Write the ratio.

b cos(θ) = AH

2 Substitute the values of the angle and the pronumerals into the cosine ratio.

cos(70°) = y5

3 Make y the subject of the equation. y = 5 cos(70°)4 Evaluate y using a calculator. = 1.7105 Round the answer to 2 decimal places. ≈ 1.71 km6 Answer the question in sentence form. Benjamin has travelled 1.71 km

north of the base camp.

Exercise 6.3 Calculating unknown side lengthsINDIVIDUAL PATHWAYS

⬛ PRACTISEQuestions:1–3, 4a–c, 5a–c, 6a–f, 7–9, 11, 12

⬛ CONSOLIDATEQuestions:1–3, 4b–d, 5b–d, 6d–i, 7–10, 12–15

⬛ MASTERQuestions:1–3, 4d–f, 5d–f, 6g–i, 7–10, 11, 13–17

FLUENCY

1 WE4 a Evaluate the following correct to 4 decimal places.i sin(55°) ii sin(11.6°)

b Copy and complete the table below. Use your calculator to find each value of sin(θ) correct to 2 decimal places.

θ 0° 15° 30° 45° 60° 75° 90°sin(θ)

c Summarise the trend in these values.2 a Evaluate the following correct to 4 decimal places.

i cos(38°) ii cos(53.71°)

REFLECTIONWhat does sin(60°) actually mean?

⬛ ⬛ ⬛ Individual pathway interactivity int-4499

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186 Maths Quest 9


b Copy and complete the table below. Use your calculator to find each value of cos(θ) correct to 2 decimal places.

θ 0° 15° 30° 45° 60° 75° 90°cos(θ)

c Summarise the trend in these values.3 a Evaluate the following correct to 4 decimal places.

i tan(18°) ii tan(51.9°)b Copy and complete the table below. Use your calculator to find each value of tan(θ)

correct to 2 decimal places.

θ 0° 15° 30° 45° 60° 75° 90°tan(θ)

c Find the value of tan(89°) and tan(89.9°).d What do you notice about these results?

4 WE5 Use the appropriate trigonometric ratios to find the length of the unknown side in each of the triangles shown. Give the answers correct to 2 decimal places.a

x

50°

17 m

b

y

27°7.9 m

c

31°46 mm

z

d p 78° 13.4 cm e

z

12°

29.5 m

f s

22°37.8 m

5 WE6 Use the appropriate trigonometric ratio to find the length of the unknown side in each of the triangles shown. Give the answers correct to 2 decimal places.a

k

75°

11 cm

b 52°

16 cm

s

c 5°

16.1 cm

q

d 5.72 km66°

p

e

7.7 km

24°e f

29.52 m

t72°

eles-0116


Trigonometry 187

6 Find the length of the unknown side in each of the following triangles, correct to 2 decimal places. (Note: In some cases the unknown will be in the numerator and in other cases it will be in the denominator.)a

119.83 mm

l

13°

b

46.7 cma39°

c

0.95 km

x

62°

d

1.73 km

y

33° e 39.75 m

d

8° f 40.25 m

y75°

g

63.2 mm

30°

h

98 cm

n

67° i z

17°312.18 mm

7 Find the lengths of the unknown sides in the triangles shown, correct to 2 decimal places.a

a

b17.8°

58.73

b

a c

b30° 40°

18 c

c ab

42° 63°

38

UNDERSTANDING

8 MC a The value of x correct to 2 decimal places is:A 59.65 B 23.31 C 64.80 D 27.51

b The value of x correct to 2 decimal places is:A 99.24 mm B 92.55 mm C 185.55 mm D 198.97 mm

c The value of y correct to 2 decimal places is:A 47.19 B 7.94 C 1.37 D 0.23

d The value of y correct to 2 decimal places is:A 0.76 km B 1.79 kmC 3.83 km D 3.47 km

x

67° 25.32

x

47°135.7 mm

y

3.386°

y

1.62 km

25°

188 Maths Quest 9


9 WE7 A ship that 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 diagram.a How far east had the ship travelled?b How far north had the ship travelled?

10 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?

11 Walking along the coastline, Michelle (M) looks up through an angle of 55° and sees her friend Helen (H) on top of the cliff at the lookout point. How high is the cliff if Michelle is 200 m from its base? (Assume both girls are the same height.)

200 m

H

M55°

80° 280 km

N

E

27°19 m

192 Maths Quest 9


Finding the angle when 2 sides are knownIf the lengths of any 2 sides of a right-angled triangle are known, it is possible to find an angle using inverse sine, inverse cosine or inverse tangent.

Determine the value of θ in the triangle at right. Give your answer correct to the nearest degree.

THINK WRITE/DRAW

1 Label the given sides. These are used in CAH. Write the ratio.

63Hypotenuse

Adjacent12

θ

cos(θ) = AH

2 Substitute the given values into the cosine ratio. cos(θ) = 1263

3 θ is the inverse cosine of 1263

. θ = cos−1 a1263b

4 Evaluate. = 79.0

5 Round the answer to the nearest degree. ≈ 79°

Roberta enjoys water skiing and is about to try a new ramp on the Hawkesbury River. The inclined ramp rises 1.5 m above the water level and spans a horizontal distance of 6.4 m. What is the magnitude (size) of the angle that the ramp makes with the water? Give the answer correct to the nearest degree.

THINK WRITE/DRAW

1 Draw a simple diagram, showing the known lengths and the angle to be found.

6.4Adjacent

Opposite1.5θ

2 Label the given sides. These are used in TOA. Write the ratio.

tan(θ) = OA

3 Substitute the values of the pronumerals into the tangent ratio.

tan(θ) = 1.56.4

4 θ is the inverse inverse tangent of 1.56.4

. θ = tan−1a1.56.4b

WORKED EXAMPLE 10

63

12θ

WORKED EXAMPLE 11

6.4 m

1.5 m

Trigonometry 193


5 Evaluate. = 13.19

6 Round the answer to the nearest degree. ≈ 13°7 Write the answer in words. The ramp makes an angle of 13°

with the water.

Exercise 6.4 Calculating unknown anglesINDIVIDUAL PATHWAYS

⬛ PRACTISEQuestions:1a–c, 2, 3a–f, 4–10

⬛ CONSOLIDATEQuestions:1d–f, 2, 3d–h, 4–11

⬛ MASTERQuestions:1g–i, 2, 3e–i, 4–13

FLUENCY

1 WE8 Evaluate each of the following, correct to the nearest degree.a sin−1(0.6294) b cos−1(0.3110) c tan−1(0.7409)d tan−1(1.3061) e sin−1(0.9357) f cos−1(0.3275)g cos−1(0.1928) h tan−1(4.1966) i sin−1(0.2554)

2 WE9 Determine the size of the angle in each of the following. Give answers correct to the nearest degree.a sin(θ) = 0.3214 b sin(θ) = 0.6752 c sin(β) = 0.8235d cos(β) = 0.9351 e cos(α) = 0.6529 f cos(α) = 0.1722g tan(θ) = 0.7065 h tan(a) = 1 i tan(b) = 0.876j sin(c) = 0.3936 k cos(θ) = 0.5241 l tan(α) = 5.6214

3 WE10 Determine the value of θ in each of the following triangles. Give answers correct to the nearest degree.a

72

49θ

b

8560

θ c 16.54

15.16

θ

d 41.32 38.75

θ

e

12.61

12.61

θ f 6.9

21.8

θ

g 26

28.95

θ h 21.72

105.62

θ

i

76.38

78.57

θ

REFLECTIONWhy does cos(0°) = 1?


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194 Maths Quest 9


4 MC a If cos(θ) = 0.8752, the value of θ correct to 2 decimal places is:A 61.07° B 41.19° C 25.84° D 28.93°

b If sin(θ) = 0.5530, the value of θ correct to 2 decimal places is:A 56.43° B 33.57° C 28.94° D 36.87°

c The value of θ in the triangle shown, correct to 2 decimal places, is:A 41.30°B 28.55°C 48.70°D 61.45°

d The value of θ in the triangle shown, correct to 2 decimal places, is:A 42.10°B 64.63°C 25.37°D 47.90°

5 a Copy and fill in the table below.

x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y = cos−1 (x) 90° 60° 0°b Plot the above table on graph paper or with a spreadsheet or suitable calculator.

UNDERSTANDING

6 A piece of fabric measuring 2.54 m by 1.5 m has a design consisting of parallel diagonal stripes. What angle does each diagonal make with the length of the fabric? Give your answer correct to 2 decimal places.

7 WE11 Danny Dingo is perched on top of a cliff 20 m high watching an emu feeding 8 m from the base of the cliff. Danny has purchased a flying contraption, which he hopes will help him capture the emu. At what angle to the cliff must he swoop to catch his prey? Give your answer correct to 2 decimal places.

20 m

8 m

119.65136.21

θ

785.2

709.5

θ

1.5 m

2.54 m

Trigonometry 197


6.5 Angles of elevation and depression When looking up towards an object, an angle of elevation

is the angle between the horizontal line and the line of vision.

When looking down at an object, an angle of depression is the angle between the horizontal line and the line of vision.

Angles of elevation and depression are measured from horizontal lines.

At a point 10 m from the base of a tree, the angle of elevation of the treetop is 38°. How tall is the tree to the nearest centimetre?

THINK WRITE/DRAW

1 Draw a simple diagram. The angle of elevation is 38° from the horizontal.

38°10

Adjacent

Oppositeh

2 Label the given sides of the triangle. These sides are used in TOA. Write the ratio.

tan(38°) = h10

3 Multiply both sides by 10. 10 tan(38°) = h4 Calculate correct to 3 decimal places. h = 7.812

5 Round to 2 decimal places. ≈ 7.81

6 Write the answer in words. The tree is 7.81 m tall.

A lighthouse, 30 m tall, is built on top of a cliff that is 180 m high. Find the angle of depression (θ) of a ship from the top of the lighthouse if the ship is 3700 m from the bottom of the cliff.

180 m

3700 m

30 mAngle of depression θ

Line of vision

Angle of elevation

Horizontal

Line of vision

Horizontal

Angle of depression

WORKED EXAMPLE 12

WORKED EXAMPLE 13

198 Maths Quest 9


REFLECTIONWhy does the angle of elevation have the same value as the angle of depression?

THINK WRITE/DRAW

1 Draw a simple diagram to represent the situation. The height of the triangle is 180 + 30 = 210 m. Draw a horizontal line from the top of the triangle and mark the angle of depression, θ. Also mark the alternate angle.

Opposite210

CS

T

3700Adjacent

θ

θ

2 Label the triangle. These sides are used in TOA. Write the ratio.

tan(θ) = OA

3 Substitute the given values into the ratio. tan(θ) = 2103700

4 θ is the inverse tangent of 2103700

. θ = tan−1a 2103700

b

5 Evaluate. = 3.2

6 Round the answer to the nearest degree. ≈ 3°7 Write the answer in words. The angle of depression of

the ship from the top of the lighthouse is 3°.

Note: In Worked example 13, the angle of depression from the top of the lighthouse to the ship is equal to the angle of elevation from the ship to the top of the lighthouse. This is because the angle of depression and the angle of elevation are alternate (or ‘Z’) angles.

This can be generalised as follows: For any two objects, A and B, the angle of elevation of B,

as seen from A, is equal to the angle of depression of A, as seen from B.

Angle of depressionof A from B

Angle of elevationof B from A

A

B

θ

θ

Exercise 6.5 Angles of elevation and depressionINDIVIDUAL PATHWAYS

⬛ PRACTISEQuestions:1–6, 8, 10, 12

⬛ CONSOLIDATEQuestions:1–6, 8, 11–14

⬛ MASTERQuestions:1–3, 6, 7, 9–16

Angle of depression

Angle ofelevation

θ

θ


Trigonometry 199


FLUENCY

1 WE12 Building specifications require the angle of elevation of any ramp constructed for public use to be less than 3°.

Ramps being constructed at a new shopping centre are each made in the ratio 7 m horizontal length to 1 m vertical height. Find the angle of elevation of these ramps and, hence, decide whether they meet building specifications.

2 A lifesaver standing on his tower 3 m above the ground spots a swimmer experiencing difficulty. The angle of depression of the swimmer from the lifesaver is 12°. How far is the swimmer from the lifesaver’s tower? (Give your answer correct to 2 decimal places.)

3 From the top of a lookout 50 m above the ground, the angle of depression of a camp site that is level with the base of the lookout is 37°. How far is the camp site from the base of the lookout?

UNDERSTANDING

4 From a rescue helicopter 80 m above the ocean, the angles of depression of two shipwreck survivors are 40° and 60° respectively. If the two sailors and the helicopter are in line with each other:a draw a labelled diagram to represent the situationb calculate the distance between the two sailors, to the nearest metre.

5 The angle of elevation of the top of a tree from a point on the ground, 60 m from the tree, is 35°.a Draw a labelled diagram to represent the situation.b Find the height of the tree to the nearest metre.

1 m

7 m

12°

3 m

50 m

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6.2 what is trigonometry? the hypotenuse is opposite the right angle. adjacent hypotenuse! opposite...

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