Trigonometry and Heights & Distances Quiz: SSC Practice

Spoilers! These are the Starter quiz’s answers. Easy, Medium, Hard and Mixed rounds show their own answers when you finish.

1. What is the value of sin²60° + cos²45°?

   • 1
   • 5/4
   • 3/2
   • 7/4

   Answer: 5/4. sin 60° = sqrt(3)/2 so sin²60° = 3/4; cos 45° = 1/sqrt(2) so cos²45° = 1/2. Total 5/4.

2. If sin A = 3/5 and A is acute, what is tan A?

   • 3/4
   • 4/3
   • 4/5
   • 5/3

   Answer: 3/4. Opposite 3, hypotenuse 5, so adjacent = 4 (3-4-5 triangle). tan A = 3/4.

3. If sec A + tan A = 3, what is the value of sec A − tan A?

   • 3
   • 2/3
   • 1/3
   • 1/9

   Answer: 1/3. (sec A + tan A)(sec A − tan A) = sec²A − tan²A = 1, so sec A − tan A = 1/3.

4. If sin 3A = cos (A − 10°), where 3A is acute, what is A?

   • 20°
   • 25°
   • 30°
   • 22.5°

   Answer: 25°. sin x = cos y with acute angles means x + y = 90°: 3A + A − 10° = 90°, so A = 25°.

5. What is the value of tan 1° × tan 2° × tan 3° × ... × tan 89°?

   • 0
   • 89
   • 1/2
   • 1

   Answer: 1. Pair tan A with tan (90° − A) = cot A; each pair multiplies to 1, and the middle term tan 45° = 1.

6. When the sun's elevation is 60°, a tower casts a shadow 20 m long. What is the height of the tower?

   • 20 sqrt(3) m
   • 20/sqrt(3) m
   • 40 m
   • 10 sqrt(3) m

   Answer: 20 sqrt(3) m. tan 60° = height/shadow, so height = 20 × sqrt(3) = 20 sqrt(3) m (about 34.6 m).

7. A 10 m ladder leans against a wall and makes an angle of 60° with the ground. How high up the wall does it reach?

   • 5 m
   • 10 sqrt(3) m
   • 5 sqrt(3) m
   • 5 sqrt(2) m

   Answer: 5 sqrt(3) m. Height = ladder × sin 60° = 10 × sqrt(3)/2 = 5 sqrt(3) m.

8. The angle of elevation of the top of a tower from a point on the ground is 30°. On walking 60 m towards the tower, it becomes 60°. What is the height of the tower?

   • 20 sqrt(3) m
   • 60 m
   • 30 m
   • 30 sqrt(3) m

   Answer: 30 sqrt(3) m. h cot 30° − h cot 60° = 60 gives h(sqrt(3) − 1/sqrt(3)) = 60, so h × 2/sqrt(3) = 60 and h = 30 sqrt(3) m.

9. What is the maximum value of 5 sin A + 12 cos A?

   • 17
   • 12
   • 13
   • 7

   Answer: 13. a sin A + b cos A has maximum sqrt(a² + b²) = sqrt(25 + 144) = 13.

10. From the top of a cliff 50 m high, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff?

    • 50 m
    • 50 sqrt(3) m
    • 50/sqrt(3) m
    • 100 m

    Answer: 50 sqrt(3) m. The angle of depression equals the angle of elevation from the boat. tan 30° = 50/d, so d = 50 sqrt(3) m (about 86.6 m).