Year 9 Mathematics Trigonometry Practice Test 4

Name__________________________

1 Label the sides of the triangle below hypotenuse, opposite and adjacent

2 Label the triangle below with opposite, hypotenuse and θ

3 For this triangle, write down the expressions for the sine, cosine and tangent ratios of the given angle

4 Write the trigonometric ratio which must be used in order to find the value of the pronumeral in each of the following triangles.

a)

C h a p t e r 1 4 R i g h t - a n g l e d t r i g o n o m e t r y 495Trigonometric ratios are used to determine the unknown side lengths and angles of aright-angled triangle.

To determine which one of the three trigonometric ratios to use in each case, followthese steps.Step 1 Label the sides of the triangle, which are either given, or need to be found,

using the symbols O, A, H with respect to the angle in question.Step 2 Consider the sides which are involved and write the trigonometric ratio con-

taining these sides. (Use the mnemonic SOH–CAH–TOA to assist you.)Step 3 Identify the values of the pronumerals in the ratio.Step 4 Substitute the values of the pronumerals into the ratio.

Write the trigonometric ratio which must be used in order to find the value of the pronumeral in each of the following triangles.a b

THINK WRITE

a Label the sides of the triangle whose lengths are given, using the appropriate symbols.

a

We are given the lengths of the opposite side (O) and the hypotenuse (H). Write down the ratio that contains both of these sides.

sin ! =

Identify the values of the pronumerals. O = 6, H = 15Substitute the values of the pronumerals into the ratio. (Since the given angle is denoted with the letter b, replace ! with b.)

sin b =

b Label the sides of the triangle whose lengths are either given, or need to be found, using the appropriate symbols.

b

The length of the adjacent side (A) is given and the length of the opposite side (O) needs to be found. Write down the ratio that contains these sides.

tan ! =

Identify the values of the pronumerals. O = x, A = 18, ! = 50° Substitute the values of the pronumerals into the ratio.

tan 50° =

615

b

50°

18

x

1

6 = O15 = H

b

2OH----

3

4615------

1

50°

18 = Ax = O

2 OA----

3

4x

18------

5WORKEDExample

MQ 9 NSW 5.2P - 14 Page 495 Sunday, July 25, 2004 9:08 AM

b)

C h a p t e r 1 4 R i g h t - a n g l e d t r i g o n o m e t r y 495Trigonometric ratios are used to determine the unknown side lengths and angles of aright-angled triangle.

To determine which one of the three trigonometric ratios to use in each case, followthese steps.Step 1 Label the sides of the triangle, which are either given, or need to be found,

using the symbols O, A, H with respect to the angle in question.Step 2 Consider the sides which are involved and write the trigonometric ratio con-

taining these sides. (Use the mnemonic SOH–CAH–TOA to assist you.)Step 3 Identify the values of the pronumerals in the ratio.Step 4 Substitute the values of the pronumerals into the ratio.

Write the trigonometric ratio which must be used in order to find the value of the pronumeral in each of the following triangles.a b

THINK WRITE

a Label the sides of the triangle whose lengths are given, using the appropriate symbols.

a

We are given the lengths of the opposite side (O) and the hypotenuse (H). Write down the ratio that contains both of these sides.

sin ! =

Identify the values of the pronumerals. O = 6, H = 15Substitute the values of the pronumerals into the ratio. (Since the given angle is denoted with the letter b, replace ! with b.)

sin b =

b Label the sides of the triangle whose lengths are either given, or need to be found, using the appropriate symbols.

b

The length of the adjacent side (A) is given and the length of the opposite side (O) needs to be found. Write down the ratio that contains these sides.

tan ! =

Identify the values of the pronumerals. O = x, A = 18, ! = 50° Substitute the values of the pronumerals into the ratio.

tan 50° =

615

b

50°

18

x

1

6 = O15 = H

b

2OH----

3

4615------

1

50°

18 = Ax = O

2 OA----

3

4x

18------

5WORKEDExample

MQ 9 NSW 5.2P - 14 Page 495 Sunday, July 25, 2004 9:08 AM

5 Evaluate each of the following giving answers correct to 4 decimal places

a) sin 56° b) cos 35° c) tan 78°

6 Find the value of x correct to 2 decimal places

7 Find the value of m correct to 2 decimal places7

C h a p t e r 1 4 R i g h t - a n g l e d t r i g o n o m e t r y

501

In the previous example we found the opposite side, having been given the adjacent side.After substituting into the tangent ratio formula, the

x

was in the numerator and the givenside was in the denominator. In the next worked example the unknown side is in thedenominator.

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/DISPLAY

Label the sides of the triangle using the symbols O, A, H with respect to the 58° angle.

We are finding the opposite side (O), given the adjacent side (A). Write the ratio containing these sides.

tan ! =

Identify the values of the variables. ! = 58°, O = x, A = 16.2 mSubstitute the appropriate values (the values of !, O and H) into the tangent ratio.

tan 58° =

Make x the subject of the equation by multiplying both sides by 16.2.

x = 16.2 " tan 58°

Use a calculator to evaluate, then round your answer to 2 decimal places.On a graphics calculator, press 16.2 " 58 .

x = 25.925 419 37x # 25.93 m

1

x = O

H

58°

16.2 m = A

2OA----

34

x16.2----------

5

6

TAN EXE

7WORKEDExample

x58°

16.2 m

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

Continued over page

THINK WRITE/DISPLAYLabel the sides of the triangle using the symbols O, A, H with respect to the 22° angle.

We are finding the hypotenuse (H) given the adjacent side (A). Write the ratio containing these sides.

cos ! =

Identify the values of the variables. ! = 22°, A = 17.4 cm, H = m

117.4 cm = A

m = H

O

22°

2AH----

3

8WORKEDExample17.4 cm

m22°

MQ 9 NSW 5.2P - 14 Page 501 Wednesday, July 28, 2004 7:16 AM

8 Calculate each of the following correct to 2 decimal places

9 Find the size of angle ! correct to the nearest degree, given sin ! = 0.6583.

10 Find the value of !:"

a) correct to the nearest minute, given that cos ! = 0.2547

b) correct to the nearest second, given that tan ! = 2.364.

11 For each of the following find the size of the angle marked with a pronumeral, correct to the nearest degree.

12 Find the size of angle θ to the nearest minute in each triangle below.

a) cos 65° 57’ b) tan 56° 45’ 30”

a)

510 M a t h s Q u e s t 9 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y

For each of the following find the size of the angle marked with a pronumeral, correct to the nearest degree.a b

THINK WRITE/DISPLAYa Label the sides of the triangle. a

Identify the appropriate trigonometric ratio to use. We are given O and H, so choose the sine ratio.

sin ! =

Substitute O = 3.5 and H = 5 and evaluate the expression.

sin ! =

= 0.7Make ! the subject of the equation using inverse sine.

! = sin"1(0.7)

Evaluate ! and round, correct to the nearest degree.

! # 44°

b Label the sides of the triangle. b

Identify the appropriate trigonometric ratio to use. We are given O and A, so choose the tangent ratio.

tan ! =

Substitute O = 5 and A = 11. As the value of tan ! is a simple fraction, we do not need to evaluate the expression.

tan ! =

5 cm3.5 cm

!

11 m

5 m

!

1

3.5 cm5 cm

!

H O

A

2OH----

33.55

-------

4

5

1

5 m

11 m!

HO

A

2OA----

35

11------

12WORKEDExample

MQ 9 NSW 5.2P - 14 Page 510 Sunday, July 25, 2004 9:08 AM

b)

510 M a t h s Q u e s t 9 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y

For each of the following find the size of the angle marked with a pronumeral, correct to the nearest degree.a b

THINK WRITE/DISPLAYa Label the sides of the triangle. a

Identify the appropriate trigonometric ratio to use. We are given O and H, so choose the sine ratio.

sin ! =

Substitute O = 3.5 and H = 5 and evaluate the expression.

sin ! =

= 0.7Make ! the subject of the equation using inverse sine.

! = sin"1(0.7)

Evaluate ! and round, correct to the nearest degree.

! # 44°

b Label the sides of the triangle. b

Identify the appropriate trigonometric ratio to use. We are given O and A, so choose the tangent ratio.

tan ! =

Substitute O = 5 and A = 11. As the value of tan ! is a simple fraction, we do not need to evaluate the expression.

tan ! =

5 cm3.5 cm

!

11 m

5 m

!

1

3.5 cm5 cm

!

H O

A

2OH----

33.55

-------

4

5

1

5 m

11 m!

HO

A

2OA----

35

11------

12WORKEDExample

MQ 9 NSW 5.2P - 14 Page 510 Sunday, July 25, 2004 9:08 AM

a)

C h a p t e r 1 4 R i g h t - a n g l e d t r i g o n o m e t r y 511

When asked for a more accurate measurement of an angle we are able to use the calcu-lator to find an angle correct to the nearest minute or nearest second.

THINK WRITE/DISPLAY

Make ! the subject of the equation using inverse tangent.

! = tan"1( )

Evaluate ! and round, correct to the nearest degree.

! # 24°

On a Casio graphics calculator, this would be as shown on the screen at right.

4511------

5

Find the size of angle ! in each of the triangles shown below.a b

Continued over page

THINK WRITE/DISPLAYa Label the sides of the triangle. a

Identify the appropriate trigonometric ratio to use.

tan ! =

(Answer correct to the nearest minute.)

3.1 m

7.2 m

!

(Answer correct to the nearest second.)

55 cm

42 cm!

1 3.1 m

7.2 m

!

H

O

A

2OA----

13WORKEDExample

MQ 9 NSW 5.2P - 14 Page 511 Sunday, July 25, 2004 9:08 AM

b)

C h a p t e r 1 4 R i g h t - a n g l e d t r i g o n o m e t r y 511

When asked for a more accurate measurement of an angle we are able to use the calcu-lator to find an angle correct to the nearest minute or nearest second.

THINK WRITE/DISPLAY

Make ! the subject of the equation using inverse tangent.

! = tan"1( )

Evaluate ! and round, correct to the nearest degree.

! # 24°

On a Casio graphics calculator, this would be as shown on the screen at right.

4511------

5

Find the size of angle ! in each of the triangles shown below.a b

Continued over page

THINK WRITE/DISPLAYa Label the sides of the triangle. a

Identify the appropriate trigonometric ratio to use.

tan ! =

(Answer correct to the nearest minute.)

3.1 m

7.2 m

!

(Answer correct to the nearest second.)

55 cm

42 cm!

1 3.1 m

7.2 m

!

H

O

A

2OA----

13WORKEDExample

MQ 9 NSW 5.2P - 14 Page 511 Sunday, July 25, 2004 9:08 AM

13 A ladder of length 3 m makes an angle of 32° with the wall.

a) How far is the foot of the ladder from the wall?

"b) How far up the wall does the ladder reach?"c What angle does the ladder make with the ground?

14 From an observer, the angle of elevation of the top of a tree is 50°. If the observer is 8 metres from the tree, find the height of the tree.

15 Change each of the following compass bearings to true bearings.

16 Change each of the following true bearings to compass bearings. a 220°T b 130°T

17 A boat travels a distance of 5 km from A to B in a direction of 035°T.

a) How far east of A is B?

b) How far north of A is B?

c) What is the true bearing of A from B?

a) N32°E

b) S14°W

a) 220°T

b) 130°T