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TRIGONOMETRIC EQUATIONS

What is a Trigonometric equation?

A trigonometric equation is an equation involving trigonometric functions of functions of unknown angles. The solution of a trigonometric equation is a value of the unknown angle that satisfies the equation. A trigonometric equation can have unlimited number of solutions, eg. If sin x=0, then x can have values 0, p,2p,3p…. The solution to the equation which has a value lying between 0 and 2p is called the principal solution. Since a trigonometric function is periodic, a solution generalised by the means of periodicity is known as the general solution. Thus , we can say, that every trigonometric equation has a principal value as well as a general solution.

NOTE:

1. In addition remember that sin x and cos x only take values between -1 and 1 so the equations sin x = 2 or sin x = -5 do not have any solution or rather are mathematically incorrect.
2. The reciprocals of sin x and cos x, namely, sec x and cosec x take values in the interval (-

P, - 1] U [1, P). So sec x = 1/2, cosec x = -3/4 etc. have no solution.

3. Cot x and tan x take all real values.

SOLUTIONS TO SOME STANDARD TRIGONOMETRIC EQUATIONS

• sin x = a

è x = np + (-1)ⁿsin^-1a

• cos x = a

è x = 2np +- cos^-1a

• tan x = a

è x = np + tan^-1a

• sin x = 0

è x = np

• cos x = o

è x = (2n + 1) p/2

• tan x = 0

è x = np

• sin x = 1

è x = (4n + 1) p/2

• cos x = 1

è x = 2np

• tan x = 1

è x = np + p/4

• sin x = -1

è x = (4n -1)p/2

• cos x = -1

è x = (2n+1) p

• tan x = -1

è x = np - p/4

where n is any integer.