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

$A$ tower of height $h$' standing at the centre of a square with sides of length $a$' makes the same angle $\alpha$ at each of the four corners then $a^{2}=$
  1. $ h^{2}\cot^{2}\alpha$
  2. $ h^{2}\tan^{2}\alpha$
  3. $ 2h^{2}\cot^{2}\alpha$
  4. $ 2h^{2}\tan^{2}\alpha$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the tower be at the center (0,0,0) with height h. The corners of the square are at (+-a/2, +-a/2, 0). The distance from the center to a corner is sqrt((a/2)^2 + (a/2)^2) = sqrt(a^2/4 + a^2/4) = a/sqrt(2). The angle alpha is formed by the tower height h and the distance from the base to the corner. So tan(alpha) = h / (a/sqrt(2)) = h * sqrt(2) / a. Thus, a = h * sqrt(2) / tan(alpha) = h * sqrt(2) * cot(alpha). Squaring both sides: a^2 = 2 * h^2 * cot^2(alpha).

Multiple choice

The angle of elevation of an electric pole from a point to the ground is $60^{ o}$ and from a point $B$ towards the pole on the line joining the foot of the pole to the point is $75^{o}$. If the distance $AB=a$, then the height of the pole is:
  1. $\displaystyle \frac{a(3+2\sqrt{3})}{2}$
  2. $a(4+2\sqrt{3})$
  3. $\displaystyle \frac{a(2+\sqrt{3})}{2}$
  4. $\displaystyle \frac{a(2\sqrt{3}-3)}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let height be h. Distance from pole = h/tan(60) = h/sqrt(3). Distance from B = h/tan(75). a = h/tan(60) - h/tan(75) = h(1/sqrt(3) - (2-sqrt(3))). Solving for h gives a(3+2sqrt(3))/2.

Multiple choice

The angle of elevation of the top of an unfinished tower from a point at a distance of $120$ metres from its base is $45^{0}$. The height of the tower that must be raised so that the angle of elevation at the same point will be $60^{0}$ is
  1. $120(\sqrt{3}-1)m$
  2. $120(\sqrt{3}+1)m$
  3. 120 $\sqrt{3}m$
  4. 140 $\sqrt{3}m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Initial height h1 = 120 * tan(45) = 120. Final height h2 = 120 * tan(60) = 120 * sqrt(3). Height to be raised = 120 * sqrt(3) - 120 = 120(sqrt(3) - 1).

Multiple choice

At a certain point the angle of elevation of a tower is found to be $\cot^{-1}\dfrac35$. On walking $32\ m$ directly towards the tower its angle of elevation is $\cot^{-1}\dfrac25$ . The height of the tower (in metres) is
  1. $32$
  2. $160$
  3. $320$
  4. $340$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let h be height, x be initial distance. cot(theta1) = x/h = 3/5, so x = 3h/5. After walking 32m, distance is x-32. cot(theta2) = (x-32)/h = 2/5, so x-32 = 2h/5. Substituting x: 3h/5 - 2h/5 = 32. h/5 = 32, so h = 160.

Multiple choice

Semivertical angle of a cone of maximum volume of given slant height is
  1. $\tan ^{ -1 }{ 2 } $
  2. $\cot ^{ -1 }{ 2 } $
  3. $\tan ^{ -1 }{ \sqrt { 2 } } $
  4. $\cot ^{ -1 }{ \sqrt { 2 } } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a cone of fixed slant height l, the volume V = (1/3) * pi * r^2 * h. With r = l * sin(theta) and h = l * cos(theta), V = (1/3) * pi * l^3 * sin^2(theta) * cos(theta). Maximizing this leads to tan(theta) = sqrt(2).

Multiple choice

Find the maximum perimeter of a triangle on a given base '$a$' and having the given vertical angle $\displaystyle \alpha$
  1. $\displaystyle P_{max}=a\left ( 1+\mathrm{cosec}\frac{\alpha }{2} \right )$
  2. $\displaystyle P_{max}=2a\left ( 1+\cos\frac{\alpha }{2} \right )$
  3. $\displaystyle P_{max}=2a\left ( 1+\mathrm{cosec}\frac{\alpha }{2} \right )$
  4. $\displaystyle P_{max}=a\left ( 1+\cos\frac{\alpha }{2} \right )$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The maximum perimeter of a triangle with a fixed base and vertical angle is achieved when the triangle is isosceles.

Multiple choice

A rocket of height $\displaystyle h$ meters is fired vertically upwards.Its velocity at time t seconds is $\displaystyle (2t+3)$ meters.second.If the angle of elevation of the top of the rocket from a point on the ground after 1 second of firing is $\displaystyle \pi /6$ and after 3 seconds it is $\displaystyle \pi /3$ then the distance of the point from the rocket is :
  1. $\displaystyle 14\sqrt{3} $ meters
  2. $\displaystyle 7\sqrt{3} $ meters
  3. $\displaystyle 2\sqrt{3} $ meters

  4. cannot be found without the value of h
Reveal answer Fill a bubble to check yourself
B Correct answer
Multiple choice

The sine of the angle between the vectors $\hat{i}+3\hat{j}+2\hat{k}$ and $2\hat{i}-4\hat{j}+\hat{k}$ is
  1. $\sqrt{\dfrac{155}{156}}$
  2. $\sqrt{\dfrac{115}{116}}$
  3. $\sqrt{\dfrac{115}{147}}$
  4. $\sqrt{\dfrac{157}{158}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let u = (1, 3, 2) and v = (2, -4, 1). The cross product u x v = (3*1 - 2*-4, 2*2 - 1*1, 1*-4 - 3*2) = (11, 3, -10). Magnitude |u x v| = sqrt(121 + 9 + 100) = sqrt(230). Magnitudes |u| = sqrt(1+9+4) = sqrt(14), |v| = sqrt(4+16+1) = sqrt(21). sin(theta) = |u x v| / (|u||v|) = sqrt(230) / sqrt(14*21) = sqrt(230/294) = sqrt(115/147).

Multiple choice

The angle between the lines whose cosines are $\left(\dfrac{-\sqrt{3}}{4},\dfrac{1}{4},\dfrac{-\sqrt{3}}{2}\right)$ and $\left(\dfrac{-\sqrt{3}}{4},\dfrac{1}{4},\dfrac{-\sqrt{3}}{2}\right)$, is given by ___________.
  1. $30^o$
  2. $0^o$
  3. $45^o$
  4. $60^o$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The angle between two lines with direction cosines (l1, m1, n1) and (l2, m2, n2) is given by cos(theta) = l1l2 + m1m2 + n1n2. Since the vectors are identical, the dot product is 1, so cos(theta) = 1, meaning theta = 0 degrees.

Multiple choice

Find the range of $\displaystyle f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}-\frac{\cos x}{\sqrt{1+\cot^{2}x}}$
  1. $[0,1]$
  2. $[-1,0]$
  3. $[-2,2]$
  4. $\{0\}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Simplify the expression: sin(x)/sqrt(1+tan^2(x)) = sin(x)/|sec(x)| = sin(x)cos(x)/|cos(x)|. If cos(x) > 0, this is sin(x)cos(x)/cos(x) = sin(x). If cos(x) < 0, this is -sin(x). Similarly, cos(x)/sqrt(1+cot^2(x)) = cos(x)/|csc(x)| = cos(x)sin(x)/|sin(x)|. The expression simplifies to 0 for all x where defined.

Multiple choice

If $\theta$ is an acute angle such that $\sec^2 \theta = 3$. then the value of $\dfrac{\tan^2 \theta - \text{cosec}^2 \theta}{\tan^2 \theta - \text{cosec}^2 \theta}$ is
  1. $\dfrac{4}{7}$
  2. $\dfrac{3}{7}$
  3. $1$
  4. $\dfrac{1}{7}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expression is (tan^2(theta) - cosec^2(theta)) / (tan^2(theta) - cosec^2(theta)). Since the numerator and denominator are identical, the expression simplifies to 1, provided the denominator is not zero. Given sec^2(theta) = 3, tan^2(theta) = sec^2(theta) - 1 = 2, and cosec^2(theta) = 1 + cot^2(theta) = 1 + 1/2 = 1.5, the denominator is 2 - 1.5 = 0.5, which is non-zero.

Multiple choice

Find: $\dfrac{{\cos {{45}^0}}} {{\sec {{30}^0} + \text{cosec}{{30}^0}}}$
  1. $\dfrac{2\sqrt 6}{\sqrt3 +1}$
  2. $\dfrac{\sqrt 2}{2\sqrt3 +2}$
  3. $\dfrac{2\sqrt 3}{\sqrt3 +1}$
  4.  $ \dfrac{\sqrt{3}}{2\left( \sqrt{2}+\sqrt{6} \right)} $ 
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

cos 45 = 1/sqrt(2). sec 30 = 2/sqrt(3). cosec 30 = 2. Denominator = 2/sqrt(3) + 2 = (2 + 2sqrt(3))/sqrt(3). Expression = (1/sqrt(2)) / ((2 + 2sqrt(3))/sqrt(3)) = sqrt(3) / (sqrt(2) * 2 * (1 + sqrt(3))) = sqrt(3) / (2 * (sqrt(2) + sqrt(6))).

Multiple choice

If $\displaystyle \tan \theta = \frac{1}{\sqrt7}$ and $\theta$ is an acute angle, find $\displaystyle \frac{\text{cosec}^2 \theta - \sec^2 \theta}{\text{cosec}^2 \theta + \sec^2 \theta}$.
  1. $\displaystyle \frac{3}{4}$
  2. $\displaystyle \frac{1}{2}$
  3. $2$
  4. $\displaystyle \frac{5}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given tan(theta) = 1/sqrt(7), then sec^2(theta) = 1 + tan^2(theta) = 1 + 1/7 = 8/7. cosec^2(theta) = 1 + cot^2(theta) = 1 + 7 = 8. Substituting these into the expression: (8 - 8/7) / (8 + 8/7) = (48/7) / (64/7) = 48/64 = 3/4.

Multiple choice

Choose the correct answer from the alternatives given : What is the value of $sin^3 60^\circ \, cot 30^\circ \, - \, 2 sec^2 45^\circ \, + 3 cos 60^\circ \, tan 45^\circ \, - tan^2 60^\circ$
  1. $\dfrac{35}8$
  2. $\dfrac{-35}8$
  3. $\dfrac{-11}8$
  4. $\dfrac{11}8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Substitute the values: (sqrt(3)/2)^3 * sqrt(3) - 2*(sqrt(2))^2 + 3*(1/2)*1 - (sqrt(3))^2. This simplifies to (3*sqrt(3)/8)*sqrt(3) - 4 + 1.5 - 3 = (9/8) - 4 + 1.5 - 3 = 1.125 - 5.5 = -4.375, which is -35/8.