Quantitative Aptitude

Trigonometry

435 Questions
All Height and Distance 268 Triangle Properties and Rules 50 Standard Angle Evaluation 33 Complementary and Odd-Even Angles 30 Value of one ratio from another 26 Trigonometric Identities 14 Maxima and Minima 6 General and Principal Solutions 5 Quadratic Substitution 3

Trigonometry Questions

Multiple choice

Which information are necessary to answer the questions asked- What is the height of the tower? A. It is viewed at a distance of 48 m at an angle of elevation of 30°. B. It casts a shadow of 25 m. C. If it is viewed at a certain distance it makes an angle of elevation of 30°, and when it moves 23 m towards the tower the angle of elevation becomes 60°.

  1. Only A and B together

  2. Any two of them

  3. Only C

  4. B and either A or C

  5. Questions can’t be answered even after using all the informations.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Statement A alone gives height = 48 × tan 30°. Statement C gives two angles at known distances, allowing tan relationships. Statement B (shadow) needs the sun's elevation angle, which either A or C can provide.

Multiple choice

If, θ is an acute angle and tanθ + cotθ =2 Then find the value of sec2θ -1? / यदि, θ एक न्यून कोण है और tan θ+ cot θ = 2 है, तो sec2θ -1 का मान ज्ञात कीजिये?

  1. 0

  2. 1

  3. 2

  4. √2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given tanθ + cotθ = 2. Since cotθ = 1/tanθ, let tanθ = t. Then t + 1/t = 2, giving t² - 2t + 1 = 0, so (t-1)² = 0. Thus tanθ = 1. The identity sec²θ - 1 = tan²θ gives sec²θ - 1 = 1² = 1. Option B is correct.

Multiple choice

In ΔABC, ∠A is a right angle and AD is perpendicular to BC. If AD = 3 cm, BC = 12 cm then the value of (cot B + cot

  1. is-

  2. 36

  3. 4

  4. 3

  5. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In right triangle ABC with right angle at A, AD is the altitude to hypotenuse BC. Using the relationship that in a right triangle, the product of altitude and hypotenuse equals the product of the two legs: AD × BC = AB × AC. Also, cot B = BD/AD and cot C = CD/AD. Since BD + CD = BC, we get cot B + cot C = BC/AD = 12/3 = 4. The claimed answer B (4) is correct.

Multiple choice

A pole is broken at a point above the ground due to storm. Its top just touches the ground at a distance of 10√3 m from its foot and makes an angle of 30o with the horizontal. Then height of the pole is:

  1. 20

  2. 24

  3. 25

  4. 30
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The broken pole forms a right triangle where: (1) the broken top touches ground 10√3 m away, (2) makes 30° with horizontal, (3) the standing portion is vertical. Let standing portion = h. Then tan 30° = h/(10√3) = 1/√3, giving h = 10. The broken portion = (10√3)/cos 30° = 10√3/(√3/2) = 20. Total pole height = 10 + 20 = 30m. Answer D is correct.

Multiple choice

The angles of elevation of the top of a tower from two points on the ground at the distance 32 m and 18 m from its base and in the same straight line with it are complementary. The height (in m) of the tower is:

  1. 20

  2. 24

  3. 16

  4. 28
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let angles be θ and 90°-θ. Then h/32=tan(θ) and h/18=tan(90°-θ)=cot(θ). Multiplying: (h/32)(h/18)=tan(θ)·cot(θ)=1, so h²=576 and h=24m. Complementary angles have tangents that are reciprocals.

Multiple choice

The angle of depression of two points P and Q on the ground on either side of pole, from top of the poles are 60° and 30° respectively. Then find the height (in meter) of the pole if the distance between point P and Q is 84√3m?

  1. 60

  2. 52.5

  3. 73.5

  4. 63
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the pole height be h. For point P (angle 60°): tan(60°) = h/x₁, so x₁ = h/√3. For point Q (angle 30°): tan(30°) = h/x₂, so x₂ = h√3. Total distance PQ = x₁ + x₂ = h/√3 + h√3 = 4h/√3. Given PQ = 84√3, so 4h/√3 = 84√3, giving h = (84√3 × √3)/4 = 84 × 3/4 = 63m.

Multiple choice

What is the value of [sin(90°-
    • cos(180°-2A)]/[cos(90° - 2A) + sin(180°- A)]?

  2. Sin(A/2).cosA

  3. Cot(A/2)

  4. Tan(A/2)

  5. sinA.cos(A/2)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using reduction formulas: sin(90°-A) = cosA, cos(180°-2A) = -cos2A, cos(90°-2A) = sin2A, sin(180°-A) = sinA. Expression = (cosA - cos2A)/(sin2A + sinA). Using sum-to-product: cosA - cos2A = 2sin(3A/2)sin(A/2) and sin2A + sinA = 2sin(3A/2)cos(A/2). Ratio = sin(A/2)/cos(A/2) = tan(A/2). Options A, B, and D are incorrect results.

Multiple choice

What is the value of [1 - tan(90° - θ) + sec(90° - θ)]/[tan(90° - θ) + sec(90° - θ) + 1]?

  1. Cot(θ /2)

  2. Tan(θ /2)

  3. Sin θ

  4. Cos θ
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using complementary angle identities: tan(90° - θ) = cot θ, sec(90° - θ) = cosec θ. Expression becomes [1 - cot θ + cosec θ]/[cot θ + cosec θ + 1]. Using cosec² θ - cot² θ = 1, this simplifies to [cosec θ - cot θ]/[cosec θ + cot θ]. Writing in terms of half angles: this equals tan(θ/2).

Multiple choice

If the distance between the tops of two buildings - 38 metres and 58 metres in height is 52 metres. Then find the distance (in metres) between two buildings?

  1. 46

  2. 42

  3. 44

  4. 48
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Building heights are 38 m and 58 m. Difference in height = 58 - 38 = 20 m. Distance between tops = 52 m (this is the hypotenuse). Using Pythagoras theorem: horizontal distance² + vertical difference² = hypotenuse². So d² + 20² = 52². Therefore d² = 2704 - 400 = 2304, giving d = 48 m.

Multiple choice

The angle of elevation of the top of a tower at a distance of 30m from its foot is 45°. The height of the tower is:-

  1. 20m

  2. 30m

  3. 15√2m

  4. 15/√2m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a right triangle formed by the tower, the distance from its foot, and the line of sight: tan(45°) = height/30. Since tan(45°) = 1, we get height = 30m. The 45° angle of elevation creates an isosceles right triangle where the height equals the horizontal distance. Option B is correct.

Multiple choice

$Find the value of ( \cot 25^{\circ} \cot 35^{\circ} \cot 45^{\circ} \cot 55^{\circ} \cot 65^{\circ} )$

  1. 1

  2. 0

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Use complementary angle identity: cot(θ) = tan(90°-θ). So cot 25° = tan 65°, cot 35° = tan 55°. The expression becomes tan 65° × tan 55° × cot 45° × cot 55° × cot 65°. Since cot 45° = 1 and tan θ × cot θ = 1, we get tan 65° × cot 65° × tan 55° × cot 55° × 1 = 1 × 1 × 1 = 1.

Multiple choice

(\text{If } \sin 3A = \cos(30^\circ - A), \text{ where A is an acute angle, then what is the value of } 2\csc A + \tan^2 2A - \cot^2 A)
  1. $\(\frac{17}{2}\)$
  2. $\(4\)$
  3. $\(5\)$
  4. $\(\frac{7}{4}\)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using complementary angle identity sin θ = cos(90° - θ), we have sin 3A = cos(90° - 3A). Equating with cos(30° - A): 90° - 3A = 30° - A (valid for acute angles). Solving: 60° = 2A, so A = 30°. Then 2csc 30° + tan²(60°) - cot²(30°) = 2(2) + (√3)² - (√3)² = 4 + 3 - 3 = 4.

Multiple choice

(\text{If } 2\sin(3x - 15)° = 1, 0° < (3x - 15) < 90°, \text{ then find the value of } \cos^2(2x + 15)° + \cot^2 (x + 15)°)
  1. $\(\frac{7}{2}\)$
  2. $\(\frac{5}{25}\)$
  3. $\(1\)$
  4. $\(-\frac{7}{2}\)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

2sin(3x-15) = 1, so sin(3x-15) = 1/2. Since 0° < (3x-15) < 90°, we have (3x-15) = 30°, so 3x = 45°, x = 15°. Then cos²(2x+15) = cos²(45°) = (1/√2)² = 1/2. And cot²(x+15) = cot²(30°) = (√3)² = 3. Sum = 1/2 + 3 = 7/2. The key is solving for x using the principal value of sine in the first quadrant.