3. (10pts) solve for x ∈ 0,2π): tan(x) sin(x) - √3 sin(x)= 0 a) x ∈ {π/6, 5π/6, 7π/6, 11π/6} b) x ∈…

3. (10pts) solve for x ∈ 0,2π): tan(x) sin(x) - √3 sin(x)= 0 a) x ∈ {π/6, 5π/6, 7π/6, 11π/6} b) x ∈ {0,π,π/3, 4π/3} c) x ∈ {π/4, 3π/4,π/6, 5π/6} d) x ∈ {0,π/2,π, 3π/2}

3. (10pts) solve for x ∈ 0,2π): tan(x) sin(x) - √3 sin(x)= 0 a) x ∈ {π/6, 5π/6, 7π/6, 11π/6} b) x ∈ {0,π,π/3, 4π/3} c) x ∈ {π/4, 3π/4,π/6, 5π/6} d) x ∈ {0,π/2,π, 3π/2}

Answer

Explanation:

Step1: Factor out sin(x)

$\sin(x)(\tan(x)-\sqrt{3}) = 0$

Step2: Set each factor equal to zero

Case 1: $\sin(x)=0$. In the interval $[0, 2\pi)$, when $\sin(x)=0$, $x = 0$ or $x=\pi$. Case 2: $\tan(x)=\sqrt{3}$. In the interval $[0, 2\pi)$, when $\tan(x)=\sqrt{3}$, $x=\frac{\pi}{3}$ or $x = \frac{4\pi}{3}$.

Answer:

B. $x\in\left{0,\pi,\frac{\pi}{3},\frac{4\pi}{3}\right}$