# equation .Find x if sin^2x=1/4 .

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find x if (sin x)^2 = 1/4

(sin x)^2 = 1/4

=> sin x = 1/2 and sin x = -1/2

x = arc sin (1/2) and x = arc sin (-1/2)

arc sin (1/2) = 30 degrees and arc sin (-1/2) = -30 degrees.

As the sin function is periodic all values of x after an interval of 360 degrees give the same value for sin x

This gives the result as x = 30 + n*360 and x = -30 + n*360

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll re-write the equation multiplying by 4 both sides:

4(sin x)^2 = 1

4(sin x)^2 - 1 = 0

We'll re-write the difference of two squares as a product:

4(sin x)^2 - 1 = (2 sin x - 1)(2 sin x + 1)

We'll cancel the product:

(2 sin x - 1)(2 sin x + 1) = 0

We'll put each factor as zero:

2 sin x - 1 = 0

2 sin x = 1

We'll divide by 2:

sin x = 1/2

x = (-1)^k*arcsin (1/2) + k*pi

x = (-1)^k*pi/6 + k*pi

We'll put the other factor as zero:

2 sin x + 1 = 0

We'll subtract 1 both sides:

2 sin x = -1

sin x = -1/2

x = (-1)^(k+1)*pi/6 + k*pi

The solutions of the equation are: {(-1)^k*pi/6 + k*pi}U{(-1)^(k+1)*pi/6 + k*pi}.