Trig identities!

Question

Please help, I have no idea! 

 

Solve the following trigonometric equation sin^2x - cos^2x = 1/4. 

Answer 1

sin²(x) - cos²(x) = 1/4

 

 

Subtract 1/4 on both sides of equation.

 

sin²(x) - cos²(x) - (1/4) = 0

 

 

Use the identity  sin²(x) + cos²(x) = 1

 

 

sin²(x) - (1 - sin²(x)) - (1/4) = 0

 

sin²(x) - 1 + sin²(x) - (1/4) = 0

 

2sin²(x) - 1 - (1/4) = 0

 

2sin²(x) - 5/4 = 0

 

2sin²(x) = 5/4

 

sin²(x) = 5/8

 

sin(x) = ±√(5) / ±√(8)

 

sin(x) = ±√(5) / 2√(2)

 

sin(x) = ±(2√(10) / 8)

 

sin(x) = ± √(10) / 4

 

x = 52.24

 

Sine is positive in the 1st and 2nd quadrant.

 

x = 180 - 52.24

x = 127.76

 

The solutions are

 

x = 52.24

x = 127.76