Double-Angle Identities Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Solve

\sin(2x) = \cos(x)

for

x \in [0, 2\pi)

Solution

1
Expand: $2\sin(x)\cos(x) = \cos(x)$ . Rearrange: $2\sin(x)\cos(x) - \cos(x) = 0$ , so $\cos(x)(2\sin(x) - 1) = 0$ .
2
$\cos(x) = 0$ gives $x = \frac{\pi}{2}, \frac{3\pi}{2}$ . $\sin(x) = \frac{1}{2}$ gives $x = \frac{\pi}{6}, \frac{5\pi}{6}$ .

Answer

x = \frac{\pi}{6},\; \frac{\pi}{2},\; \frac{5\pi}{6},\; \frac{3\pi}{2}

Applying the double-angle formula converts the equation to a factorable form. Setting each factor to zero and solving gives all solutions. Always check the full interval and remember not to divide by a trig function that could be zero.

About Double-Angle Identities

Formulas expressing $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ in terms of single-angle trig functions.

Learn more about Double-Angle Identities →

More Double-Angle Identities Examples

Example 1 easy

If [formula], find [formula] using the double-angle formula.

Example 2 medium

Find [formula] given that [formula] and [formula] is in Quadrant I.

Example 3 medium

Simplify [formula] using a double-angle identity.

← All Double-Angle Identities Examples Double-Angle Identities Concept

Related Concepts

Sum and Difference Identities