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TRIGONOMETRIC EQUATIONS

Introduction:

An equation involving one or more trigonometrical ratios of unknown angle is called a trigonometric equation. e.g.,


` cos x =1/2; sin^2 x -4 cos x=1`


The value of an unknown angle which satisfies the given trigonometric equation is called a solution or
root of the equation. For example `2 sin theta = sqrt 3 ` , Clearly ` theta= 60^(circ)` and `120^(circ)` are solutions of the equation
between `0^(circ)` and `360^(circ)` .


Now suppose `tan theta = 1`, then there will be many possible vlaues of `theta`, so our main objective is to write
down all the solution in short form. Since all trigonometric functions are periodic and therefore solution of
all trigonometrical equation can be generalized with the help of periodicity of trigonometrical function.

Principal solution of a Trigonometric Equation :

The solutions of a trigonometric equation `sin theta =1/2 ` lying in the interval `[0, 2 pi] ` are `pi/6` and `(5 pi)/6` .Thus principal solutions of

` sin theta =1/2 ` will be `pi/6` and `(5 pi)/6`

General solution of a Trigonometric Equation :

(i) If `sin theta = sin alpha => theta = n pi+ (-1)^n alpha`, where ` alpha in [-pi/2 , pi/2 ] , n in I`.

(ii) If `cos theta = cos alpha => theta = 2 n pi pm alpha`, where ` alpha in [0, pi], n on I`

(iii) If `tan theta = tan alpha => theta = n pi + alpha ` , where `alpha in (-pi/2, pi/2), n in I`

(iv) If `sin^2 theta = sin^2 alpha => theta = n pi pm alpha `

(v) lf `cos^2 theta = cos^2 alpha => n pi pm alpha`

(vi) lf `tan^2 theta = tan^2 alpha => theta = n pi pm alpha`


Note : `alpha` is called the principal angle.

Proof :

(i) `sin theta = sin alpha => sin theta - sin alpha =0 => 2 cos ((theta+alpha)/2) sin ((theta - alpha)/2) =0`
`=> cos ((theta+alpha)/2) =0 ` or `sin ( (theta - alpha)/2)=0`

`=> (theta+ alpha)/2 =(2m+1) pi/2 ` or `(theta - alpha)/2 = m pi`, where `m in I`
`=> theta = (2m +1) pi - alpha ` or `theta = 2m pi + alpha` , where `m in I`

`=> theta = (2m+1) pi +(-1)^(2m+1) alpha ` or `theta =2 m pi+ (-1)^2m alpha`
`=> theta = n pi + (-1)^(n) alpha , n in I`



(ii) `cos theta = cos alpha=> cos alpha - cos theta =0 => 2 sin ((alpha + theta)/2) sin ((theta - alpha)/2) =0`

`=> sin (theta + alpha)/2 =0 ` or `sin (theta -alpha)/2 =0 , (theta+ alpha)/2 = n pi` or `(theta - alpha) /2 = n pi`

`=> theta = 2 n pi -alpha ` or `theta = 2 n pi+ alpha n in I`

`=> theta = 2 n pi pm alpha`

(iii) `tan theta = tan alpha => sin theta/cos theta = sin alpha /cos alpha => sin theta cos theta- cos theta * sin alpha =0`

`=> sin (theta -alpha) =0 => theta - alpha = n pi`
`=> theta = n pi+ alpha ` , where `n in I`


(iv) `sin^2 theta = sin^2 alpha `

`sin^2 theta - sin^2 alpha = sin (theta+ alpha) sin (theta - alpha) =0`

`sin (theta+ alpha )=0` or `sin (theta -alpha)=0`

`theta+ alpha = n pi ` or `theta - alpha = n pi, n in I`

`theta = n pi pm alpha , n in I`

(v) `cos ^2 theta = cos^2 alpha => - sin^2 theta -1 -sin^2 alpha => sin^2 theta - sin^2 alpha`

`theta = n pi pm alpha , n in I`


(vi) `tan^2 theta = tan^2 alpha => tan theta = pm tan alpha = tan(pm alpha)`

`=> theta = n pi pm alpha ` , where `n in I`
 
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