Trigonometric Equations Questions with Solutions

14 questions4 easy7 medium3 hard

A trigonometric equation is an equation involving trigonometric functions of an unknown angle, such as 2 sin x = 1. This topic covers finding principal solutions (usually between 0 and 2 pi) and general solutions, which include all possible angles. The key general solutions are: sin x = sin a gives x = n pi + (-1)^n a, cos x = cos a gives x = 2n pi +/- a, and tan x = tan a gives x = n pi + a, where n is any integer. Exam questions involve reducing an equation to one of these standard forms, often by factorising or using identities, and then counting the number of solutions in a given interval. Always check that solutions do not make any term in the original equation undefined. Practise the trigonometric equations questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Consider the graphs of the functions $f (x) = 2 cos (\frac{x}{2}) + 3$ and $g (x) = 4$ . The number of points of intersection of the two graphs in the interval $[0, 4 π]$ is:
A. 4B. 3C. 1D. 2
easyAsked in NIMCET 2026View solution ›
Q2.
If $s in (π cos θ) = cos (π s in θ)$ , then $s in 2 θ$ is equal to
A. $\pm \frac{3}{4}$ B. $\pm \frac{1}{3}$ C. $\pm \frac{1}{4}$ D. $\pm \frac{4}{3}$
hardAsked in NIMCET 2012View solution ›
Q3.
The general solution of $3 cos x + s in x = 3$ is
A. $2 nπ \pm \frac{π}{6}$ B. $2 nπ \pm \frac{π}{3}$ C. No solutionD. $nπ \pm \frac{π}{6}$
easyAsked in NIMCET 2011View solution ›
Q4.
The general solution of $2 s in^{2} θ - 3 s in θ - 2 = 0$ is
A. $nπ + (- 1)^{n} \frac{π}{6}$ B. $nπ + (- 1)^{n} \frac{π}{2}$ C. $nπ + (- 1)^{n} \frac{5 π}{6}$ D. $nπ + (- 1)^{n} \frac{7 π}{6}$
mediumAsked in NIMCET 2011View solution ›
Q5.
The equation $s in^{4} x + cos^{4} x + s in 2 x + α = 0$ is solvable for:
A. $- \frac{1}{2} \leq α \leq \frac{1}{2}$ B. $- 3 \leq α \leq 1$ C. $- \frac{3}{2} \leq α \leq \frac{1}{2}$ D. $- 1 \leq α \leq 1$
mediumAsked in NIMCET 2009View solution ›
Q6.
The general value of $θ$ satisfying $s in θ = - \frac{1}{2}$ and $t an θ = \frac{1}{3}$ is
A. $nπ + \frac{π}{6}, n \in I$ B. $nπ + (- 1)^{n} \frac{7 π}{6}, n \in I$ C. $2 nπ + \frac{7 π}{6}, n \in I$ D. $2 nπ + \frac{11 π}{6}, n \in I$
mediumAsked in NIMCET 2021View solution ›
Q7.
If $32 t an^{8} θ = 2 cos^{2} α - 3 cos α$ and $3 cos 2 θ = 1$ , then the general value of $α$ is
A. $nπ \pm \frac{π}{3}$ B. $2 nπ \pm \frac{2 π}{3}$ C. $2 nπ \pm \frac{π}{3}$ D. $nπ \pm \frac{2 π}{3}$
mediumAsked in NIMCET 2021View solution ›
Q8.
The solutions of the equation $4 cos^{2} x + 6 s in^{2} x = 5$ are
A. $x = nπ \pm \frac{π}{4}$ B. $x = nπ \pm \frac{π}{3}$ C. $x = nπ \pm \frac{π}{2}$ D. $x = nπ \pm \frac{2 π}{3}$
easyAsked in NIMCET 2022View solution ›
Q9.
The value of A that satisfies the equation $a s in A + b cos A = c$ is equal to
A. $t an^{- 1} (\frac{a}{b}) \pm cos^{- 1} (\frac{c}{a ^{2} + b ^{2}})$ B. $t an^{- 1} (\frac{c}{b}) \pm cos^{- 1} (\frac{a}{a ^{2} + b ^{2}})$ C. $t an^{- 1} (\frac{a}{b}) \pm s in^{- 1} (\frac{c}{a ^{2} + b ^{2}})$ D. None of these
hardAsked in NIMCET 2017View solution ›
Q10.
The number of solutions of the equation $s in x + s in 5 x = s in 3 x$ lying in the interval $[0, π]$ is
A. 4B. 6C. 5D. 2
mediumAsked in NIMCET 2018View solution ›
Q11.
Let $P = θ : s in θ - cos θ = 2 cos θ$ and $Q = θ : s in θ + cos θ = 2 s in θ$ be two sets. Then
A. $P \subset Q$ and $Q - P \neq = \emptyset$ B. $P ⊄ Q$ C. $Q ⊄ P$ D. $P = Q$
mediumAsked in NIMCET 2018View solution ›
Q12.
If $s in^{2} x t an x + cos^{2} x co t x - s in 2 x = 1 + t an x + co t x$ , $x \in (0, π)$ , then x =
A. $\frac{3 π}{12}$ , $\frac{5 π}{12}$ B. $\frac{5 π}{12}$ , $\frac{7 π}{12}$ C. $\frac{7 π}{12}$ , $\frac{11 π}{12}$ D. $\frac{7 π}{12}$ , $\frac{9 π}{12}$
hardAsked in NIMCET 2019View solution ›
Q13.
Find the cardinality of the set $C = x : s in 4 x = \frac{1}{2}, x \in (- 9 π, 3 π)$ .
A. 24B. 48C. 36D. 12
mediumAsked in NIMCET 2024View solution ›
Q14.
What is the general solution of the equation $t an θ + co tθ = 2$ ?
A. $θ = nπ + \frac{π}{4}$ B. $θ = \frac{nπ}{2} + \frac{π}{8}$ C. $θ = \frac{nπ}{2} + \frac{π}{6}$ D. $θ = \frac{nπ}{2} + \frac{π}{4}$
easyAsked in NIMCET 2025View solution ›