Trig Equations

Solve Using Trig Identity

Use the Pythagorean identity sin²x + cos²x = 1 to reduce a trig equation, then solve the simplified form on [0, 2π).

solve-with-identity Strategy: recognize 1 - sin²x = cos²x (Pythagorean identity), substitute to get an equation in cos only, then solve. cos²x = 0 ⟹ cos x = 0 ⟹ x = π/2, 3π/2 on [0, 2π).

Example

Use a trig identity to reduce the equation before solving.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

1−sin⁡2x=01 -\sin^{ 2 }x= 0

Step 2 — Use identity

Use the Pythagorean identity to replace 1 minus sine squared.

1−sin⁡2x=cos⁡2x1 -\sin^{ 2 }x=\cos^{ \hl{2} }x

Step 3 — Reduce

Reduce the equation to cosine squared equals 0.

cos⁡2x=0\cos^{ \hl{2} }x= 0

Step 4 — Solution set

Cosine is zero at two angles in the interval.

x∈{π2,3π2}x\in\{ \hlmath{\frac{\pi}{2}} , \hlmath{\frac{3\pi}{2}} \}