Equations involving trigonometric functions of unknown angles are called trigonometric equations. Numerically least angle is called the principal value (solution).

Since, trigonometric functions are periodic, a solution is generalised by means of periodicity of the function. The solution consisting of all possible solutions of a trigonometric equation is called its general solution .

Method of an auxiliary angle

Equations of the form $\mathrm{a} \cos \theta \pm \mathrm{b} \sin \theta=\mathrm{c}$ are equivalent to the elementary trigonometric equations

$\cos (\theta \mp \phi)=\dfrac{\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$ where $\sin \phi=\dfrac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}, \cos \phi=\dfrac{\mathrm{a}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$

If $|c|>\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}$, then $\mathrm{a} \cos \theta \pm \mathrm{b} \sin \theta=\mathrm{c}$ has no solution.

If $|c| \leq \sqrt{a^{2}+b^{2}}$, then put

$\dfrac{|\mathrm{c}|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}=\cos \alpha$ so that

$\cos (\theta \mp \phi)=\cos \alpha$

$\Rightarrow \quad \theta \mp \phi=2 \mathrm{n} \pi \pm \alpha$

$\Rightarrow \quad \theta=2 \mathrm{n} \pi \pm \alpha \pm \phi ; \quad \mathrm{n} \in \mathrm{Z}$

Note :

(i) While solving, if you are squaring a trigonometric equation (which should be avoided as far as possible), check the solution for extraneous roots.

(ii) Do not cancel terms containing unknown terms which are in product (it may cause loss of a valid solution)

(iii) Remove all those values (angles) which make any of the terms undefined or infinite.

Solving trigonometric equations graphically

Sketch the graph of each side of the equation. Now look for all the points of intersection within the given interval.

Solved Examples

1. The number of distinct solutions of

$\sin 5 \theta \cos 3 \theta=\sin 9 \theta \cos 7 \theta \text { in }\left[0, \dfrac{\pi}{2}\right] \text { is }$

(a) 4

(b) 5

(c) 8

(d) 9

2. The number of values of $x$ in $[0,5 \pi]$ satisfying $3 \cos 2 x-10 \cos x+7=0$ is

(a) 5

(b) 6

(c) 8

(d) 10

Show Answer

Solution:

$3\left(2 \cos ^{2} x-1\right)-10 \cos x+7=0$ $3 \cos ^{2} \mathrm{x}-5 \cos \mathrm{x}+2=0$

$\Rightarrow \cos x=\dfrac{2}{3}, 1$.

Clearly the lines $\mathrm{y}=\dfrac{2}{3}, 1$ intersect the graph at 8 points

Hence 8 values

Answer: (c)

3. If $\sin ^{2} \theta-2 \sin \theta-1=0$ is to be satisfied for exactly 4 distinct values of $\theta \in[0, n \pi], \forall n \in N$; then the least value of $n$ is

(a) 2

(b) 6

(c) 4

(d) 8

Show Answer

Solution:

$\begin{aligned} & \sin ^{2} \theta-2 \sin \theta-1=0 \\ & \sin \theta=1 \pm \sqrt{2} \\ \Rightarrow & \sin \theta=1-\sqrt{2} \text { (as } 1+\sqrt{2} \text { is rejected) }\end{aligned}$

clearly four solutions lie in $[0,4 \pi]$

$\therefore$ least value of $\mathrm{n}$ is 4

Also $5^{\text {th }}$ solution lies in $[0,5 \pi]$

$\therefore$ greatest value can be 5 .

Answer: (c)

4. The most general solution of secx $-1=(\sqrt{2}-1) \tan x$ is

(a) $\mathrm{n} \pi+\dfrac{\pi}{8}$

(b) $2 \mathrm{n} \pi, 2 \mathrm{n} \pi+\dfrac{\pi}{4}$

(c) $2 \mathrm{n} \pi$

(d) None of these

5. Sum of all the solutions of

$\cos x \cdot \cos \left(\dfrac{\pi}{3}+x\right) \cdot \cos \left(\dfrac{\pi}{3}-x\right)=\dfrac{1}{4}, x \in[0,6 \pi]$ is

(a) $15 \pi$

(b) $30 \pi$

(c) $\dfrac{110 \pi}{3}$

(d) None

of these

6. The number of solutions of $\cos x=|1+\sin x|$ for $x \in[0,3 \pi]$, is

(a) 3

(b) 2

(c) 4

(d) None of these

Show Answer

Solution:

It is evident from the figure that, the two graphs intersect at 3 points.

Answer: (a)

7. The number of solutions of $16^{\sin ^{2} x}+16^{\cos ^{2} x}=10$ in $x \in[0,2 \pi]$ is

(a) 8

(b) 6

(c) 4

(d) 2

Show Answer

Solution:

$16^{\sin ^{2} x}+16^{1-\sin ^{2} x}=10$

put $16^{\sin ^{2} x}=y \Rightarrow y+\dfrac{16}{y}=10$

$\mathrm{y}^{2}-10 \mathrm{y}+16=0 \quad$ gives $\mathrm{y}=8,2$

$\therefore 16^{\sin ^{2} x}=2,8=(16)^{\dfrac{1}{4}} \&(16)^{\dfrac{3}{4}}$

$\therefore \sin ^{2} \mathrm{x}=\dfrac{1}{4}, \dfrac{3}{4}$

Clearly lines $\mathrm{y}=\dfrac{1}{4}, \dfrac{3}{4}$ intersect the graph in 8 points

Hence 8 possible solutions.

Answer: (a)

Exercise

1. Let $\mathrm{P}=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and

$\hspace {0.8 cm}\mathrm{Q}=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets. Then

(a) $\mathrm{P} \subset \mathrm{Q}$ and $\mathrm{Q}-\mathrm{P} \neq \phi$

(b) $\mathrm{Q} \not \subset \mathrm{P}$

(c) $\mathrm{P} \not \subset \mathrm{Q}$

(d) $\mathrm{P}=\mathrm{Q}$

2. The number of values of $\theta$ in the interval $\left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$ such that $\theta \neq \dfrac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta$ is_______________

(a) 0

(b) 1

(c) 2

(d) 3

3. The positive integer value of $n>3$ satisfying the equation

$\dfrac{1}{\sin \left(\dfrac{\pi}{n}\right)}=\dfrac{1}{\sin \left(\dfrac{2 \pi}{n}\right)}+\dfrac{1}{\sin \left(\dfrac{3 \pi}{n}\right)}$ is

(a) 5

(b) 6

(c) 7

(d) 8

4.* All values of $\theta$ in the interval $\left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$ satisfying $(1-\tan \theta)(1+\tan \theta) \sec ^{2} \theta+2^{\tan ^{2} \theta}=0$

(a) $\dfrac{\pi}{3}$

(b) $\dfrac{-\pi}{3}$

(c) 0

(d) None of these.

5. Match the following :-

6. The solution set of $|4 \sin x-1|<\sqrt{5},|x|<\pi$ is

(a) $\left(-\pi,-\dfrac{4 \pi}{5}\right) \cup\left(-\dfrac{\pi}{5}, \dfrac{\pi}{10}\right) \cup\left(\dfrac{9 \pi}{10}, \pi\right)$

(b) $\left(\dfrac{-9 \pi}{10}, \dfrac{-\pi}{10}\right) \cup\left(\dfrac{3 \pi}{10}, \dfrac{7 \pi}{10}\right)$

(c) $\left(-\pi, \dfrac{-9 \pi}{10}\right) \cup\left(\dfrac{-\pi}{10}, \dfrac{3 \pi}{10}\right) \cup\left(\dfrac{7 \pi}{10}, \pi\right)$

(d) $\left(\dfrac{-7 \pi}{10}, \dfrac{9 \pi}{10}\right)$

7. The number of points $(x, y)$ inside the circle $x^{2}+y^{2}=4$ satisfying the equation $\tan ^{4} x+\cot ^{4} x+1=3 \sin ^{2} y$ is

(a) 1

(b) 2

(c) 4

(d) 8

8. If $\sin x=2 \sin \theta, \cos x=\tan y, \tan x=\cos z$ and $\cos y=\tan z$, then $\theta$ is

(a) $18^{\circ}$

(b) $36^{\circ}$

(c) $54^{\circ}$

(d) $72^{\circ}$

9. Read the following and then answer the questions.

Consider the system of equations

$\sin x \cos 2 \mathrm{y}=\left(\mathrm{a}^{2}-1\right)^{2}+1, \cos x \sin 2 \mathrm{y}=\mathrm{a}+1$

(i) the number of values of a for which the system has a solution is

(a) 1

(b) 2

(c) 3

(d) infinite.

(ii) The number of values of $\mathrm{x} \in[0,2 \pi]$ when the system has solution for permissible values of a is

(a) 1

(b) 2

(c) 3

(d) 4

(iii) The number of values of $y \in[0,2 \pi]$ when the system has solution for permissible values of a

(a) 2

(b) 3

(c) 4

(d) 5

10. The total number of solutions of $\sin {x}=\cos {x}$ in $[0,2 \pi]$ is

(a) 5

(b) 6

(c) 8

(d) None of these

11. If $\sin x+\cos x=\sqrt{\left(y+\dfrac{1}{y}\right)}, x \in[0, \pi]$ then

(a) $\mathrm{x}=\dfrac{\pi}{4}, \mathrm{y}=1$

(b) $y=2$

(c) $x=\dfrac{3 \pi}{4}$

(d) None of these.

12. If the inequatlity $\sin ^{2} x+a \cos x+a^{2}>1+\cos x$ hods for any $x \in R$ then the largest negative integral value of ’ $a$ ’ is

(a) -4

(b) -3

(c) -2

(d) -1

13. The number of solutions of $x \in[0,2 \pi]$ for which $[\sin x+\cos x]=3+[-\sin x]+[-\cos x]$ is

(a) 0

(b) 4

(c) infinite

(d) 1

14. The arithmetic mean of the roots of the equation $4 \cos ^{3} x-4 \cos ^{2} x-\cos (315 \pi+x)=1$ in the interval $(0,315)$ is

(a) $50 \pi$

(b) $51 \pi$

(c) $100 \pi$

(d) $315 \pi$

15. Values of $x$ \& y satisfying the equation

$\sin ^{7} y=\left|x^{3}-x^{2}-9 x+9\right|+\left|x^{3}-x^{2}-4 x+4\right|+\sec ^{2} 2 y+\cos ^{4} y$ are

(a) $\mathrm{x}=1, \mathrm{y}=\mathrm{n} \pi, \mathrm{n} \in \mathrm{Z}$

(b) $\mathrm{x}=1, \mathrm{y}=2 \mathrm{n} \pi+\dfrac{\pi}{2}, \mathrm{n} \in \mathrm{Z}$

(c) $x=1, y=2 n \pi, n \in Z$

(d) None of these