How to Solve Trigonometric Equations

A trigonometric equation is an equation involving the trigonometric function or function of unknown angles, e.g., sin x = 0, cos2x-sinx=1/4, sin( theta + 1/4) = ½, etc are all trigonometric equations.

A solution of a trigonometric equation is a value of that unknown angle that satisfies the equation. Trigonometric equation may have an unlimited number of solution, e.g., sin x =0,pi2pi3pi

Let us learn how to solve trigonometric equations.

how to solve the trigonometric equation step by step:

Type 1: Equation in which only one function of a single angle is involved.

Procedure. Solve algebraically for the values of the function

Solve for x, sin x =    (sqrt(-3))/(2)(0≤x≤2π)



solution:
sin x = (sqrt3)/(2)   = - sin 60°


= sin (180° + 60°)

= sin (360° - 60°)

rArr x = 240°,300°


Type 2. Equation expressible in terms of one t- ratio of the unknown angle

Procedure. The following formulas help in expressing all the t-functins in terms of a single t-function.

Square relations

Solve cos2 theta -sintheta - 1/4 = 0(0°<= theta360°)

cos2 theta -sintheta - 1/4=0

1 - sin2 theta- sintheta 1/4=0

4sin2theta +4 sintheta -3=0

hence, 2 sin  theta + 3 = 0 sin  theta

This value is inadmissible

theta = 30°,150°
Type 3 Equation involving multiple angles.

Case-I When the equation involves only one multiple angle.

solve sin 3x = - 1/sqrt(2) ,0<= <= 2pi

Solution since we require x such that 0<= <= 2pimust be such that 0<=3x <=6pi

3x = 5pi/4 ,7pi /4 , 13pi 15pi 21pi 23pi

= 5pi/4 ,7pi /4 , 13pi /4 ,15pi /4 ,21pi /4 , 23pi


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Case-2 Equation involving two multiple angles

They are like

sin ntheta = sin m theta

sin ntheta  = cos mtheta

How to solve trigonometric equations involving compound angles.

Solve the cos x - sqrt(3) sin x = 1,0°  <= x <= 0°

Solution: Dividing both sides of the equation by sqrt({(1)2) +(sqrt(3)2 }

=> cos60° cosx - sin 60°sin x = 1/2

=> cos(x + 60°)= cos60° or cos(360° - 60°)

x + 60° = 60° or 300° => x = 0°,240°

Taking values such that 0°<=  x  <= 360°.

Hence proved

The above problems says how to solve trigonometric equations.