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

From the top of a tower, the angles of depression of two objects on the same side of the tower are found to be $\alpha $ and $\beta $ where $\alpha >\beta $. The height of the tower is $130\ m,$ $\alpha =60^o: and: \beta =30^o$. The distance of the extreme object from the top of the tower is
  1. $65\ m$
  2. $130\ m$
  3. $260\ m$

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

The tower height is 130m. The angle of depression to the extreme object is beta = 30 degrees. In the right triangle formed by the tower and the ground, sin(30) = height / hypotenuse. Thus, 1/2 = 130 / hypotenuse, so the distance is 260m.

Multiple choice

The angle of depression of a point on the ground from the top of a tree is $60^{\circ}$. Lowering down 20 m from the top. the angle of depression changes to $30^{\circ}$. Find the distance between the point and the foot of the tree. Also find the height of the tree.
  1. 10 $\sqrt{3}$ ,20
  2. 20 $\sqrt{3}$ ,30
  3. 10 $\sqrt{3}$ ,30
  4. 20 $\sqrt{3}$ ,40
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let h be the height of the tree and d be the distance. tan 60 = h/d => h = d*sqrt(3). tan 30 = (h-20)/d => d*tan 30 = h-20. Substituting h: d/sqrt(3) = d*sqrt(3) - 20 => 20 = d(sqrt(3) - 1/sqrt(3)) = d(2/sqrt(3)) => d = 10*sqrt(3). h = 10*sqrt(3)*sqrt(3) = 30.

Multiple choice

The angle of elevation of the top of a tower as observed from a point on the horizontal ground is $x$. If we move a distance $d$ towards the flux of the tower, the angle of elevation increases to $y$, then the height of the tower is
  1. $\cfrac { d\tan { x } \tan { y } }{ \tan { y- } \tan { x } } $
  2. $d\left( \tan { y } +\tan { x } \right) $
  3. $d\left( \tan { y } -\tan { x } \right) $
  4. $\cfrac { 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. tan(x) = h / (d + y_dist), tan(y) = h / y_dist. y_dist = h / tan(y). h / tan(x) = d + h / tan(y). d = h(1/tan(x) - 1/tan(y)) = h(tan(y) - tan(x)) / (tan(x)tan(y)). h = d * tan(x) * tan(y) / (tan(y) - tan(x)).

Multiple choice

What is the height of a tower if the angles of elevation of its top from two points x and y at distance of a and b respectively from the base and on the same straight line with the tower, are complementary?
  1. $\sqrt b/a$
  2. $\sqrt a/b$
  3. $\sqrt {ab}$

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

Let height be h. tan(theta) = h/a and tan(90-theta) = h/b. Since tan(90-theta) = cot(theta), we have cot(theta) = h/b. Thus, tan(theta) * cot(theta) = (h/a) * (h/b) = 1. So h^2 = ab, h = sqrt(ab).

Multiple choice

The angle of elevation of the top of pole from the point $A$ on ground be $\alpha$, whereas angle of depression of the foot of pole from the point which is $'b'$ foot above point $A$ is $\beta$, then the length of pole is
  1. $\dfrac { b\cot \beta \tan \alpha  }{ \cot \beta \tan \alpha -1 }$
  2. $b \cot \alpha. \tan\beta$
  3. $b \ \tan \alpha . \tan \beta$
  4. $b \ \cot \alpha . \cot \beta$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let H be the pole height. From point A, tan(alpha) = H/d, so d = H/tan(alpha). From point b above A, tan(beta) = b/d. Substituting d, tan(beta) = b / (H/tan(alpha)) = b * tan(alpha) / H. Solving for H gives H = b * tan(alpha) / tan(beta) = b * tan(alpha) * cot(beta). The provided option A is a complex form that simplifies to this.

Multiple choice

The angles of elevation of a cliff at a point $A$ on the ground and a point $B$, $100\ m$ vertically above $A$ are $\alpha$ and $\beta$ respectively. The height of the cliff is
  1. $\dfrac{100\, cot\, \alpha}{cot\,\alpha + \cot\,\beta}$
  2. $\dfrac{100\, cot\, \beta}{cot\,\alpha - \cot\,\beta}$
  3. $\dfrac{100\, cot\, \beta}{cot\,\beta - \cot\,\alpha}$
  4. $\dfrac{100\, cot\, \beta}{cot\,\beta + \cot\,\alpha}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let height of cliff be h. From point A, cot(alpha) = distance/h. From point B, cot(beta) = distance/(h-100). Solving for h gives h = 100 * cot(beta) / (cot(beta) - cot(alpha)).

Multiple choice

At a distance a from the foot of a tower $AB$ of known height $b$ a flagstaff $BC$ and the tower subtend equal angles, height of the flagstaff is
  1. $\dfrac { a ^ { 2 } + b ^ { 2 } } { a ^ { 2 } - b ^ { 2 } }$
  2. $\dfrac { a ^ { 2 } - b ^ { 2 } } { a ^ { 2 } + b ^ { 2 } }$
  3. $\dfrac { a \left( a ^ { 2 } - b ^ { 2 } \right) } { a ^ { 2 } + b ^ { 2 } }$
  4. $\dfrac { b \left( a ^ { 2 } + b ^ { 2 } \right) } { \left( a ^ { 2 } - b ^ { 2 } \right) }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

The angle of elevation of the top of the tower observed from each of three points A, B, C on the horizontal ground, forming a triangle ABC, is observed to be $'\theta '$. If 'R' be the circumradius of the triangle, then height of tower is :
  1. $R\tan { \theta } $
  2. $R\cot { \theta } $
  3. $R\sin { \theta } $
  4. $R\cos { \theta } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If the tower's foot is O, equal angles of elevation imply OA = OB = OC, so O is the circumcenter and the horizontal distance is R. In the right triangle, tan(theta) = height/R, giving height = R tan(theta).

Multiple choice

A round balloon of radius $r$ subtends an angle $\alpha$ at the eye of an observer, while the angle of elevation of its centre is $\beta .$ Then height of the centre of the balloon is
  1. $r \sin a \ cosec \left( \dfrac { \beta } { 2 } \right)$
  2. $r \sin \beta \ cosec \left( \dfrac { \alpha} { 2 } \right)$
  3. $r \sin a \sec \left( \dfrac { \beta } { 2 } \right)$
  4. $r \sin \beta \sec \left( \dfrac { \alpha } { 2 } \right)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let O be the center of the balloon and E be the eye of the observer. The angle subtended by the balloon is alpha, so the angle between the line of sight OE and the tangent is alpha/2, giving sin(alpha/2) = r/OE, or OE = r*cosec(alpha/2). Since the angle of elevation of the center is beta, the height of the center is h = OE*sin(beta) = r*sin(beta)*cosec(alpha/2).

Multiple choice

A tree is broken at certain height and its upper part $9\sqrt2m$ long not completely separated meet the ground at an angle of ${45}^{o}$. Find the height of the tree before it was broken and also find the distance from the root of the tree to the point where the top of the tree meets the ground.
  1. $8\sqrt 2$; $11m$
  2. $7(\sqrt 2-1)$; $15m$
  3. $9(\sqrt 2+1)$; $9m$

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

Let x be the height of the broken part. The tree forms a right triangle with the ground. sin(45) = height/x => height = x*sin(45) = 9*sqrt(2) * (1/sqrt(2)) = 9. The base distance = x*cos(45) = 9. Total height = 9 + 9*sqrt(2) = 9(1+sqrt(2)).