Solving Trig Equations — Question 7

Question 7

Solve the equation $$\sin(2x) = \sqrt{3} \cos(x)$$ for all $x \in [0, 2\pi]$.

Question 7 - Solution

Use the identity: $$\sin(2x) = 2\sin(x)\cos(x)$$

Substitute into the equation: $$2\sin(x)\cos(x) = \sqrt{3}\cos(x)$$

Subtract $\sqrt{3}\cos(x)$ from both sides: $$2\sin(x)\cos(x) - \sqrt{3}\cos(x) = 0 \Rightarrow \cos(x)\left(2\sin(x) - \sqrt{3}\right) = 0$$

Now solve each factor:

**Case 1:** $\cos(x) = 0 \Rightarrow x = \frac{\pi}{2},\ \frac{3\pi}{2}$

**Case 2:** $2\sin(x) - \sqrt{3} = 0 \Rightarrow \sin(x) = \frac{\sqrt{3}}{2}$

$$\sin(x) = \frac{\sqrt{3}}{2} \Rightarrow x = \frac{\pi}{3},\ \frac{2\pi}{3}$$

Final Solution: $$\boxed{x = \frac{\pi}{3},\ \frac{2\pi}{3},\ \frac{\pi}{2},\ \frac{3\pi}{2}}$$