Trigonometric Functions & Identities Questions with Solutions

47 questions16 easy24 medium7 hard

Trigonometric functions (sine, cosine, tangent and their reciprocals) relate the angles of a right triangle to the ratios of its sides, and extend to any angle using the unit circle. This topic covers degree and radian measure, values of trigonometric ratios at standard angles, signs in the four quadrants, and allied angles such as (90 - theta) and (180 + theta). It also covers the main identities: sin^2 + cos^2 = 1 and its variations, compound angle formulas such as sin(A + B), multiple and sub-multiple angle formulas such as sin 2A and cos 2A, and the sum-to-product and product-to-sum formulas. Most questions are about choosing the right identity to simplify an expression or prove an equality, so learning these formulas thoroughly pays off. Practise the trigonometric functions and identities questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
The value of $\frac{s in ^{2} 3 θ}{s in ^{2} θ} - \frac{cos ^{2} 3 θ}{cos ^{2} θ}$ is equal to
A. $8 cos 2 θ$ B. $3 s in 2 θ$ C. $\frac{1}{8} cos 2 θ$ D. None of these
hardAsked in AMU MCA 2016View solution ›
Q2.
If $\prod_{i = 1}^{n} t an α_{i} = 1$ , where each $α_{i} \in [0, \frac{π}{2})$ , then the maximum value of $\prod_{i = 1}^{n} s in α_{i}$ is
A. $\frac{1}{2 ^{n}}$ B. $\frac{1}{2 ^{n /2}}$ C. 1D. None of these
hardAsked in NIMCET 2023View solution ›
Q3.
The value of $t an θ + 2 t an 2 θ + 4 t an 4 θ + 8 co t 8 θ$ is
A. $co tθ$ B. $t an θ$ C. $s in θ$ D. $cos θ$
hardAsked in NIMCET 2013View solution ›
Q4.
The maximum value of $(cos α_{1}) (cos α_{2}) \dots (cos α_{n})$ , where $0 \leq α_{1}, α_{2}, \dots, α_{n} \leq π$ and $(co t α_{1}) (co t α_{2}) \dots (co t α_{n}) = 1$ , is
A. $\frac{1}{2 ^{n /2}}$ B. $\frac{1}{2 ^{n}}$ C. $\frac{1}{2 n}$ D. $1$
hardAsked in NIMCET 2008View solution ›
Q5.
If $cos α + cos β = a$ , $s in α + s in β = b$ and $θ$ is the arithmetic mean between $α$ and $β$ , then $s in 2 θ + cos 2 θ$ is equal to
A. $\frac{( a + b ) ^{2}}{( a - b ) ^{2}}$ B. $\frac{( a - b ) ^{2}}{( a + b ) ^{2}}$ C. $\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}}$ D. None of these
hardAsked in NIMCET 2008View solution ›
Q6.
If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ are in arithmetic progression with common difference $d$ , then the sum $s in d (cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + \dots + cosec a_{n - 1} cosec a_{n})$ is equal to
A. $co t a_{1} - co t a_{n}$ B. $s in a_{1} - s in a_{n}$ C. $cosec a_{1} - cosec a_{n}$ D. $a_{1} - a_{n}$
hardAsked in NIMCET 2022View solution ›
Q7.
If $cos^{2} 10^{\circ} cos 20^{\circ} cos 40^{\circ} cos 50^{\circ} cos 70^{\circ} = α + \frac{3}{16} cos 10^{\circ}$ , then $3 α^{- 1}$ is equal to
A. $\frac{9}{32}$ B. 64C. 16D. $\frac{9}{16}$
hardAsked in NIMCET 2025View solution ›