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

If the angle of elevation of sun is $\displaystyle \theta $ and the length of the shadow of a pole of length is $s$, then
  1. $\displaystyle p=s\cos \theta $
  2. $\displaystyle p=s\sin \theta $
  3. $\displaystyle p=\frac{s}{\cot \theta }$
  4. $\displaystyle p=s \cot \theta $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the right triangle formed by the pole and its shadow, tan(theta) = p/s. Rearranging gives p = s tan(theta) = s/cot(theta). Thus option C is correct.

Multiple choice

Two ships are sailing in the sea on either side of a light-house, The angle of depression of the two ships are $\displaystyle 45^{\circ}$ each. if the height of the light-house is $300$ m, then the distance between ships is :
  1. $600$ m
  2. $\displaystyle 600\sqrt{3}$ m
  3. $\displaystyle 300\sqrt{3}$ m
  4. $300$ m
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The lighthouse height is 300m. The angle of depression is 45 degrees, meaning the distance from the lighthouse to each ship is 300m * cot(45) = 300m. Total distance = 300 + 300 = 600m.

Multiple choice

Two observers are stationed due north of a tower at a distance of 20 m form each other. if the elevations of the tower observed by them are $\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$ respectively, then the height of the tower is
  1. $10$ m$
  2. $16.32$ m
  3. $\displaystyle 10(\sqrt{3}+1)$ m
  4. $30$ m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the nearer observer be x m from the tower. The 45-degree elevation gives height h = x. The other observer is x+20 m away, so tan(30 degrees) = h/(h+20). Solving gives h = 10(sqrt(3)+1) m.

Multiple choice

A straight highway leads to the foot of a tower of height 50 m. From the top of the tower, angles of depressions of two cars standing on the highway are $\displaystyle 30^{\circ}$ and $\displaystyle 60^{\circ}$. what is the distance between the cars. ?
  1. $43.3 m$
  2. $57.73 m$
  3. $86.6 m$
  4. $100 m$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the tower height be h = 50. The distance of the first car is 50/tan(60) = 50/sqrt(3) and the second is 50/tan(30) = 50*sqrt(3). The distance between them is 50*(sqrt(3) - 1/sqrt(3)) = 50*(2/sqrt(3)) = 100/1.732 = 57.73 m.

Multiple choice

A tree $AC$ is broken over by over by wind from $B. D$ is the point where the top of the broken tree touches the ground and $BD$ makes an angle of $\displaystyle 45^{\circ}$ with the ground, if the distance between he base of the tree and the point $D = 10$ m. What is the height of the tree ?
  1. $20$ m
  2. $\displaystyle 10(1+\sqrt{2})$ m
  3. $\displaystyle 10\sqrt{2}$ m
  4. $\displaystyle 20\sqrt{2}$ m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the tree break at height h. The top touches the ground at distance 10m. The broken part forms the hypotenuse of a 45-45-90 triangle. Hypotenuse = 10 / cos(45) = 10 * sqrt(2). The height of the tree is the vertical part (10 * tan(45) = 10) plus the broken part (10 * sqrt(2)). Total = 10 + 10 * sqrt(2) = 10(1 + sqrt(2)).

Multiple choice

The height of a tower is h and the angle of elevation of the top of the tower is $\displaystyle \alpha $ on moving a distance h/2 towards the tower, the angle of elevation becomes. $\displaystyle \beta $. What is the value of$\quad \cot { \alpha } -\cot { \beta } \quad $
  1. $\displaystyle \dfrac{1}{2}$
  2. $\displaystyle \dfrac{2}{3}$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

From the geometry, cot(alpha) = d/h and cot(beta) = (d - h/2)/h. Subtracting these gives cot(alpha) - cot(beta) = d/h - (d/h - 1/2) = 1/2.

Multiple choice

If the angles of depression and elevation of the top of a tower of height $h$ from the top and bottom of a second tower are $x$ and $y$ respectively then the height of the second tower is
  1. $\displaystyle h\left ( \cot y+\cot x \right )$
  2. $\displaystyle h\left ( \tan x+\tan y \right )$
  3. $\displaystyle h\left ( 1+\tan x\cot y \right )$
  4. $\displaystyle h\left ( \tan y\cot x +1\right )$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the second tower have height H. The distance between towers is d. From the top of the second tower, tan(x) = h/d, so d = h/tan(x). From the bottom, tan(y) = H/d, so H = d * tan(y) = (h/tan(x)) * tan(y) = h * tan(y) * cot(x). Total height = h + H = h(1 + tan(y) * cot(x)).

Multiple choice

The angle of elevation of the top of tower as observed from a pint on the horizontal ground is 'x' if we move a distance 'd' towards the foot of the tower the angle of elevation increases to 'y' then the height of the tower is
  1. $\displaystyle \frac{d\tan x\tan y}{\tan y-\tan x}$
  2. $\displaystyle d(\tan y+\tan x)$
  3. $\displaystyle d(\tan y-\tan x)$
  4. $\displaystyle \frac{d\tan x\tan y}{\tan y+\tan x}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let h be height, y be the angle at distance x from tower, x be angle at distance x+d. tan(y) = h/x, so x = h/tan(y). tan(x) = h/(x+d). Substituting x: tan(x) = h / (h/tan(y) + d) = h*tan(y) / (h + d*tan(y)). Rearranging for h: h*tan(x) + d*tan(x)*tan(y) = h*tan(y). h(tan(y) - tan(x)) = d*tan(x)*tan(y). h = d*tan(x)*tan(y) / (tan(y) - tan(x)).

Multiple choice

A person of height $2\ m$ wants to get a fruit which is on the top of a pole of height $\displaystyle \frac{10}{3}$ m if he stands at distance of $\left (\displaystyle \frac{4}{\sqrt{3}} \right ) m$ from the foot of the pole, then the angle at which he should throw the stone, so that it hits the fruit is .......... .
  1. $15^{\circ}$
  2. $30^{\circ}$
  3. $45^{\circ}$
  4. $60^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The person's eyes are at height 2m. The fruit is at height 10/3m. The vertical distance to clear is 10/3 - 2 = 4/3m. The horizontal distance is 4/sqrt(3)m. The tangent of the angle is (4/3) / (4/sqrt(3)) = sqrt(3)/3 = 1/sqrt(3). The angle whose tangent is 1/sqrt(3) is 30 degrees.

Multiple choice

The angle of elevation of the top of a tower as seen from two points $A$ & $B$ situated the same line and at distance '$p$' and '$q$' respectively from the foot of the tower are complementary, then height of the tower is
  1. $pq$
  2. $\displaystyle \frac{p}{q}$
  3. $\displaystyle \sqrt{pq}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let height be h. tan(theta) = h/p and tan(90-theta) = h/q. Thus, cot(theta) = h/q. Since tan(theta) * cot(theta) = 1, (h/p) * (h/q) = 1, so h^2 = pq, h = sqrt(pq).

Multiple choice

The angle of elevation of the top of a tower at a distance of $\displaystyle \frac{50\sqrt{3}}{3}$ metres from the foot is $\displaystyle 60^{\circ}$. Find the height of the tower
  1. $\displaystyle 50\sqrt{3}$ metres
  2. $\displaystyle \frac{20}{\sqrt{3}}$ metres
  3. $-50$ metres
  4. $50$ metres
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Height = distance * tan(angle). Height = (50 * sqrt(3) / 3) * tan(60) = (50 * sqrt(3) / 3) * sqrt(3) = (50 * 3) / 3 = 50 meters.

Multiple choice

From the top of a light house the angles of depression of two ships on the opposite sides of it are observed to be $\alpha$ and $\beta$. If the height of the light house be h meters and the line joining the ships passes through the foot of the light house, the distance between the ships is
  1. $\dfrac {h(cot\alpha+cot\beta)}{cot\alpha . cot \beta}$
  2. $\dfrac {h(tan\alpha+tan \beta)}{tan\alpha . tan\beta}$
  3. $h(tan\alpha + tan \beta)$
  4. $\dfrac {htan\alpha . tan \beta}{tan \alpha . tan\beta}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the distance from the foot of the lighthouse to the ships be x and y. Then tan(alpha) = h/x and tan(beta) = h/y. The total distance is x + y = h/tan(alpha) + h/tan(beta) = h(cot(alpha) + cot(beta)). This is equivalent to h(tan(alpha) + tan(beta)) / (tan(alpha) * tan(beta)).

Multiple choice

The angles of elevation of the top of a tower from two points at a distance, $x$ and $y$ metres from the base and in the same straight line with it are complementary. Find the height of the tower.
  1. $h=\sqrt{xy}$
  2. $h=\dfrac{x}{y}$
  3. $h=\sqrt{y}$
  4. $h=\sqrt{x}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the height be h and the angles be theta and 90-theta. Then tan(theta) = h/x and tan(90-theta) = h/y. Since tan(90-theta) = cot(theta) = 1/tan(theta), we have h/y = x/h, which leads to h^2 = xy or h = sqrt(xy).