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 $\displaystyle \theta $ is an acute angle and $\displaystyle \tan \theta+\cot \theta =2 $ then
  1. $\displaystyle \tan \theta =\frac{1}{2} $
  2. $\displaystyle \tan \theta=2 $
  3. $\displaystyle \tan^{5} \theta+\cot ^{5}\theta=32 $
  4. $\displaystyle \tan^{7} \theta+\cot ^{7}\theta=2 $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If tan(theta) + cot(theta) = 2, then tan(theta) + 1/tan(theta) = 2. Let x = tan(theta), so x + 1/x = 2, which implies x^2 - 2x + 1 = 0, or (x - 1)^2 = 0. Thus, tan(theta) = 1, which means theta = 45 degrees. Consequently, tan^n(theta) + cot^n(theta) = 1^n + 1^n = 2 for any n.

Multiple choice

$PQ$ is a post of height $\alpha,$ and $A$ is a tower at some distance; $\alpha$ and $\beta$ are the angles of elevation of $B$, the top of tower, at $P$ and $Q$ respectively. The height of the tower is
  1. $\displaystyle \frac { a\sin { \alpha  } \cos { \beta  }  }{ \sin { \left( \alpha -\beta  \right)  }  } $
  2. $\displaystyle \frac { a\cos { \alpha  } \cos { \beta  }  }{ \sin { \left( \alpha -\beta  \right)  }  } $
  3. $\displaystyle \frac { a\sin { \alpha  } \sin { \beta  }  }{ \sin { \left( \alpha -\beta  \right)  }  } $

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

Using the tangent relation at the two points, tan(alpha) = h/x and tan(beta) = (h - a)/x. Eliminating x gives h = a sin(alpha) cos(beta) / sin(alpha - beta), which matches option A.

Multiple choice

If ABC is a triangle and $\tan\dfrac{A}{2},\tan \dfrac{B}{2},\tan \dfrac{C}{2} $ are in H.P., then find the minimum value of $\cot B/2$.
  1. $\sqrt{3}+1$
  2. $\sqrt{3}$
  3. $\sqrt{2}$
  4. $\sqrt{2}+1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a triangle, tan(A/2), tan(B/2), tan(C/2) in H.P. implies 2/tan(B/2) = 1/tan(A/2) + 1/tan(C/2). Using the identity tan(A/2)tan(B/2) + tan(B/2)tan(C/2) + tan(C/2)tan(A/2) = 1, this simplifies to tan(B/2) = 1/sqrt(3). Thus, cot(B/2) = sqrt(3).

Multiple choice

The distances of three points on a pole from its foot are in A.P. If the angles of elevations of these points from a point on the ground are $\alpha, \beta$ and $\gamma$, then $\cot\beta \cot \gamma, \cot \gamma, \cot\alpha, \cot\alpha\cot\beta$ are in?

  1. A.P.

  2. G.P.

  3. H.P.

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

Let the pole height be h and distances be x1, x2, x3 in A.P. Then cot(alpha) = x1/h, cot(beta) = x2/h, cot(gamma) = x3/h. Since x1, x2, x3 are in A.P., cot(alpha), cot(beta), cot(gamma) are in A.P. The sequence cot(beta)cot(gamma), cot(gamma)cot(alpha), cot(alpha)cot(beta) is a standard property related to A.P. terms.

Multiple choice

If the sides of a right -angled triangle are in A.P., then the sines of the acute angles are
  1. $\cfrac 3 5, \cfrac 4 5$
  2. $\cfrac {1 } {\sqrt 3}, \sqrt { \cfrac { 2 }{ 3 } } $
  3. $\cfrac { 1 }{ 2 } ,\cfrac { \sqrt { 3 } }{ 2 } $

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

If sides are in AP, they can be represented as a-d, a, a+d. For a right triangle, (a-d)^2 + a^2 = (a+d)^2. a^2 - 2ad + d^2 + a^2 = a^2 + 2ad + d^2. a^2 = 4ad => a = 4d. Sides are 3d, 4d, 5d. The sines of the acute angles are 3/5 and 4/5.

Multiple choice

If angle $A$ is acute and $\cos A= \displaystyle\frac{8}{17}$, then $\cot A$ is
  1. $\displaystyle\frac{8}{15}$
  2. $\displaystyle\frac{17}{8}$
  3. $\displaystyle\frac{15}{8}$
  4. $\displaystyle\frac{17}{15}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If cos A = 8/17, then in a right triangle, adjacent = 8 and hypotenuse = 17. Using Pythagoras, opposite = sqrt(17^2 - 8^2) = sqrt(289 - 64) = sqrt(225) = 15. Cot A = adjacent / opposite = 8/15.

Multiple choice

An aeroplane when flying at a height 2500 m from the ground passes vertically above another aeroplane At an instant when the angles of elevation of the two aeroplanes from the same point on the ground are $\displaystyle 45^{\circ}$ and $\displaystyle 30^{\circ} $ respectively then the vertical distance between the two aeroplanes at that instant is

  1. 1158.5 m

  2. 1056.5 m

  3. 1008.5 m

  4. None
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let distance from point to ground projection be x. Height of plane 1 = 2500. tan(45) = 2500/x -> x = 2500. Height of plane 2 = h. tan(30) = h/x -> h = 2500 * tan(30) = 2500 * 0.577 = 1443.5. Vertical distance = 2500 - 1443.5 = 1056.5 m.