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Solving Trigonometric Equations

Trigonometric equation is an equation that contains any of the trigonometric functions - sine, cosine, tangent, cosecent, secent and cotangent. By the solution of trigonometric equation, we mean to determine the value of the unknown angle which satisfies given trigonometric equation. The examples of trigonometric equations are:

  • $\sin \theta$ = $\frac{\sqrt{3}}{2}$
  • $\cos ^{2}\theta -3\cos \theta +2=0$

Trigonometric functions are periodic, so there is no single solution for a trigonometric equation. There are general forms of solutions of trigonometric functions sin, cos and tan. These general solutions are given below:

General Solution for Sine

General Solution of Sine

Where, n is an integer.

General Solution for Cosine

General Solution of Cosine

Where, n is an integer.

General Solution of Tangent

General Solution of Tan

Where, n is an integer.

Let us take an example:

Example: Solve $2\sin \theta =1$

Solution: $2\sin \theta =1$

$\sin \theta = $\frac{1}{2}$

$\sin \theta =\sin $$\frac{\pi }{6}$

$\theta =n\pi +(-1)^{n}$$\frac{\pi }{6}$

Where, n = 0, 1, 2, 3, .......

at n = 0,

$\theta =$$\frac{\pi }{6}$

at n = 1,

$\theta =$$\frac{5\pi }{6}$

at n = 2,

$\theta =$$\frac{13\pi }{6}$

at n = 3,

$\theta =$$\frac{17\pi }{6}$

Hence, $\theta =$$\frac{\pi }{6}$$\frac{5\pi }{6}$, $\frac{13\pi }{6}$, $\frac{17\pi }{6}$, .......

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