MCQ Questions

Question No. 1

In △ABC, find the values of each of the following.

Answer :

a) sin A

b) cos A

c) tan A

d) sin B

e) cos B

f) tan B

Question No. 2

In △DEF, find the values of each of the following.

Answer :

a) sin D

b) cos D

c) tan D

d) sin F

e) cos F

f) tan F

Question No. 3

In △GHI, find the values of each of the following.

Answer :

a) HI

b) sin H

c) cos H

d) tan H

e) sin I

f) cos I

g) tan I

Question No. 4

In △JKL, find the values of each of the following.

Answer :

a) KL

b) sin J

c) cos J

d) tan J

e) sin L

f) cos L

g) tan L

Question No. 5

In △MNO, find the values of each of the following.

Answer :

a) MO

b) sin M

c) cos M

d) tan M

e) sin O

f) cos O

g) tan O

Question No. 6

Use the trigonometric tables on pages 286 to 291 or a calculator to find each of the following trigonometric ratios. Round off your answers to two decimal places.

Answer :

a) tan 74°

b) cos 55°

c) sin 5°

d) cos 66°

e) sin 63°

f) tan 45°

g) sin 21°

h) cos 77°

Question No. 7

Use the trigonometric tables on pages 286 to 291 or a calculator to find the size of each of the following angles. Round off your answers to one decimal place.

Answer :

a) θ if cos θ = 0.43

b) θ if tan θ = 0.58

c) θ if sin θ = 0.17

d) θ if tan θ = 8.95

e) θ if cos θ = 0.71

f) θ if sin θ = 0.62

g) θ if tan θ = 2

h) θ if cos θ = 0.36

Question No. 8

Solve △PQR by calculating the values of the following.

Answer :

a) Pˆ

b) PQ

c) QR

Question No. 9

Solve △XYZ by calculating the values of the following.

Answer :

a) Xˆ

b) Zˆ

c) YZ (using trigonometry)

d) YZ (using the theorem of Pythagoras)

Question No. 10

Solve each of the triangles below by calculating the lengths of the missing sides and the sizes of the missing angles.

Answer :

a)

b)

c)

d)

Question No. 11

For each of the following triangles, first draw a rough sketch of the triangle and then solve the triangle.

Answer :

a) In △TRI, Tˆ is a right angle, TR = 6 cm and RI = 7.5 cm

b) In △HAL, Aˆ is a right angle, Hˆ = 5° and AH = 25 mm

c) In △DOG, Oˆ is a right angle, DO = 31 and GO = 58

d) In △FLY, Lˆ is a right angle, FY = 83 and Yˆ = 32°

e) In △CAT, Tˆ is a right angle, AC = 14 m and AT = 7 m

f) In △ANT, Tˆ is a right angle, Nˆ = 61° and NT = 7.3 m

g) In △PIG, Pˆ is a right angle, PI = 3.25 and GI = 5.75

h) In △HOP, Pˆ is a right angle, Oˆ = 48° and OP = 165

Question No. 12

Calculate the length of chord AB in each of the circles below.

Answer :

c)

d)

a)

b)

Question No. 13

Look at each triangle in turn. Do you understand where the side proportions come from? If not, then those members of your group who understand the triangle should help the other members to understand it as well.

Answer :

Question No. 14

Now close your textbooks. Each member of the group should try to draw these triangles for themselves. Check one another’s work and help one another to get the triangles right. It is very important to be able to draw these triangles correctly.

Answer :

Question No. 15

Use the special triangles to find the values of each of the following. Give your answers in surd form, where necessary. Do not use a calculator!

Question No. 16

Use your answers to question 3 to answer each of the following questions. You may use a calculator to convert the fractions to decimals, as these will be easier to compare.

Answer :

a) What do you notice about the values of the sine of an angle, as the angle increases from 0° to 90°?

b) What do you notice about the values of the cosine of an angle, as the angle increases from 0° to 90°?

c) What do you notice about the values of the tangent of an angle, as the angle increases from 0° to 90°?

d) What do you notice about tan 90°?

Question No. 17

Work on your own and construct a right-angled triangle with acute angles of 45° each. By measuring the appropriate sides, calculate each of the following. Round your answers off to two decimal places.

Answer :

a)(i) sin 45°

(ii) cos 45°

(iii) tan 45°

b)(i) sin 45°

Question No. 18

Work on your own and construct a right-angled triangle with acute angles of 60° and 30°. By measuring the appropriate sides, calculate each of the following. Round your answers off to two decimal places.

Answer :

a)(i) sin 60°

(ii) cos 60°

(iii) tan 60°

(iv) sin 30°

(v) cos 30°

(vi) tan 30°

b)(i) sin 60°

Question No. 19

In your groups, discuss your findings in questions 5 and 6.

Answer :

a) If some group members found answers that were not close to the correct approximate answers, help them by checking their constructions, measurements and calculations.

b) Use the special triangles to double-check the approximate values given in questions 5b) and 6b). Do you agree with these values?

Question No. 20

Simplify each of the following without using trigonometric tables or a calculator. Give your answers in surd form, where applicable.

Question No. 21

For each of the following points, calculate the values of (i) sin θ, (ii) cos θ and (iii) tan θ with the aid of a diagram, if θ is the angle between the line OP and the positive x axis.

Answer :

a) P(8; 15)

b) P(−24; 7)

c) P(6; −8)

d) P(−5; −12)

Question No. 22

Given that sin θ = −48/50 and 0° ≤ θ ≤ 270°.

Answer :

a) Draw a diagram to show this information.

b) (i) cos θ

(ii) tan θ

c) Prove by calculation that tan θ = sin θ / cos θ.

d) Prove by calculation that sin² θ + cos² θ = 1.

Question No. 23

Use a diagram to find the values of tan θ and sin θ if cos θ = 40/41 and 90° ≤ θ ≤ 360°.

Answer :

a) tan θ

b) sin θ

Question No. 24

You will need a compass and a protractor.

Answer :

a) Draw a system of axes and construct a circle with the midpoint at the origin and a radius of 10 cm.

b) Divide each quadrant of the Cartesian plane into intervals of 30°.

c) Copy the table provided; for each point on the circumference corresponding to an angle, measure its (x,y) coordinates, divide by 10 and round to two decimal places to find cos θ and sin θ.

Question No. 25

You will need a sheet of graph paper for 0° ≤ x ≤ 360°.

Answer :

a) Draw a system of axes and divide the x-axis into 30° intervals.

b) Plot the 13 sine coordinate pairs: (0°;0), (30°;0.5), …, (360°;0).

c) Join the points with a smooth curve to form the graph of y = sin x.

Question No. 26

Draw the graph of y = cos x for 0° ≤ x ≤ 360° on new graph paper.

Answer :

a) Draw axes with x divided into 30° intervals and choose a suitable y‐scale.

b) Plot the 13 cosine coordinate pairs for the same angles.

c) Join the points with a smooth curve to form the graph of y = cos x.

Question No. 27

Use your graphs and/or table to answer the following.

Answer :

a) What is the minimum value of sin x?

b) What is the maximum value of sin x?

c) What is the minimum value of cos x?

d) What is the maximum value of cos x?

Question No. 28

Use the sine and cosine graphs above to find the values of:

Question No. 29

Use the sine and cosine graphs above to find the value(s) of x for which:

Answer :

a) sin x = 0

b) sin x = 1

c) sin x = −1

d) cos x = 0

e) cos x = 1

f) cos x = −1

Question No. 30

Study the sine and cosine graphs above. Between which two values of x is:

Answer :

a) sin x > 0?

b) sin x < 0?

c) cos x < 0?

Question No. 31

A 15 m ladder is set against a wall. It makes an angle of 70° with the ground.

Answer :

a) Draw a diagram of this situation, showing all the given information.

b) How far up the wall does the ladder reach?

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

Question No. 32

This road sign says that the gradient of the slope is 4% or 2 in 50. Calculate the angle that the slope makes with the horizontal.

Answer :

a) Calculate the angle that the slope makes with the horizontal.

Question No. 33

A flagpole with a height of 7.5 m is supported by a 10 m wire that anchors the top of the pole to the ground.

Answer :

a) Draw a diagram of this situation, showing all the given information.

b) Calculate the angle between the wire and the ground.

c) Calculate the distance on the ground between the foot of the pole and the wire.

Question No. 34

A land surveyor needs to find the distance between X and Y on opposite sides of the banks of a river. He walks along the river bank from point Y and finds another point Z such that ∠XZY = 30° and YZ = 61 m. Calculate the distance between X and Y.

Answer :

a) Calculate the distance between X and Y.

Question No. 35

A tree casts a shadow of 15 metres. A beetle is on the ground at the tip of the shadow. If the angle of elevation from the beetle to the top of the tree is 29°, calculate the height of the tree, correct to the nearest centimetre.

Answer :

Question No. 36

In the diagram below, a fishing boat B is fishing off the coast of Nigeria. The boat is directly above point P, which is on the ocean bed. A fisherman F is standing on the shore, 10 km away from the boat. The angle of depression from the fisherman to point P is 4°. How far below sea level is point P? Give your answer to the nearest metre.

Answer :

Question No. 37

A game ranger is walking directly towards a vertical cliff. At point A she sees an eagle on a ledge of the cliff. The angle of elevation from the game ranger to the eagle is 22.3°. The game ranger walks directly towards the cliff for a distance of 500 m. The angle of elevation from her new position to the eagle is now 31.8°.

Answer :

a) Explain what each of the points B, C and D represent.

b) How far is point A from the foot of the cliff directly below the eagle? Give your answer to the nearest metre. (Hint: Let DC = y and BC = x, then set up two equations in terms of the given information. Solve the equations simultaneously.)

c) How high above the ground is the eagle? Give your answer to the nearest metre.

Question No. 38

In the diagram, three hikers, shown by points A, B and C, are walking in a straight line. Directly in front of them is a mast MS. The three hikers are all 10 m apart and the front hiker (at point C) is 20 m from the mast. The angle of elevation from the spot where the front hiker is to the top of the mast is 45°.

Answer :

a) What is the height of the mast?

b) Calculate the angle of depression from the top of the mast to the spot where the second hiker is.

c) Calculate the angle of elevation from the spot where the last hiker is to the top of the mast.

Question No. 39

Do this exercise as a group activity. Use the illustration of a compass while you discuss and answer the following questions. What compass point is indicated by each of these bearings?

Answer :

a) 000°

b) 270°

c) 135°

Question No. 40

Write the bearing of each of these compass points:

Answer :

a) S

b) NE

c) SW

Question No. 41

A woman and her husband go jogging. In the diagram you will see that the man jogs from their home H in a straight line on a bearing of 045°. His wife jogs from their home in a straight line on a bearing of 135°. They both jog at an average speed of 4.5 km/h. After jogging for 20 minutes, they both stop. The man is at point M and his wife is at point W. They turn and jog in a straight line towards one another at an average speed of 4.5 km/h.

Answer :

a) For how long did each of them jog altogether? Give your answer to the nearest minute.

b) How far did each of them jog in total?

Question No. 42

In the diagram alongside, Towns A, B and C form a triangle. Town B is on a bearing of 70° from Town A, Town C is on a bearing of 210° from Town B and Town A is on a bearing of 340° from Town C. The distance from Town A to Town B is 150 km.

Answer :

a) Calculate the size of ∠ABC.

b) Calculate the size of ∠ACB.

c) Calculate the size of ∠BAC.

d) Calculate the distance from Town B to Town C.

e) Calculate the distance from Town A to Town C.

Question No. 43

In the diagram below, a boat leaves a jetty J and sails 30 km on a bearing of 330°. It then turns at point A and sails on a bearing of 060° for 40 km, to point B.

Answer :

a) What is the bearing of the boat from the jetty at point B?

b) How far is the boat from the jetty at point B?

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