Introduction to Trigonometry — Chapter Test | Class 10 IMO

Q1

1 − sin²θ is equal to:

From sin²θ + cos²θ = 1, 1 − sin²θ = cos²θ.

Q2

The maximum possible value of sin θ is:

sin θ ranges from −1 to 1, so its maximum (for 0° to 90°) is 1.

Q3

The value of tan 30° is:

tan 30° = 1/√3.

Q4

The angle of depression of a boat from the top of a 75 m cliff is 45°. The boat is from the foot of the cliff:

tan 45° = 75/d = 1, so d = 75 m.

Q5

tan 90° is:

tan 90° = sin 90°/cos 90° = 1/0, which is not defined.

Q6

A ladder makes a 60° angle with the ground and its foot is 2.5 m from the wall. The ladder's length is:

cos 60° = 2.5/L, so L = 2.5 ÷ 0.5 = 5 m.

Q7

The value of sin 90° is:

sin 90° = 1.

Q8

The value of 2 sin 30° is:

2 × 1/2 = 1.

Q9

sin(90° − θ) equals:

Complementary-angle relation: sin(90° − θ) = cos θ.

Q10

The value of tan 45° is:

tan 45° = 1.

Q11

The value of cos² 45° − sin² 45° is:

Both equal 1/2, so the difference is 0.

Q12

From two points on the same side, in line with a tower's base, the angles of elevation of its top are 30° and 60°. If the points are 40 m apart, the tower's height is:

With h = d tan 60° and h = (d + 40) tan 30°, solving gives d = 20 and h = 20√3 m.

Q13

The value of cos θ when θ = 90° is:

cos 90° = 0.

Q14

The value of sin² 30° + cos² 30° is:

By the identity sin²θ + cos²θ = 1.

Q15

A ramp 12 m long rises at 30° to the horizontal. The height it reaches is:

height = 12 × sin 30° = 12 × 1/2 = 6 m.