for sin 2x + cos x = 0, use a double - angle or half - angle formula to simplify the equation and then find…

for sin 2x + cos x = 0, use a double - angle or half - angle formula to simplify the equation and then find all solutions of the equation in the interval 0, 2π). the answers are x1 =, x2 =, x3 = and x4 = with x1 < x2 < x3 < x4. question help: worked example 1 submit question jump to answer

for sin 2x + cos x = 0, use a double - angle or half - angle formula to simplify the equation and then find all solutions of the equation in the interval 0, 2π). the answers are x1 =, x2 =, x3 = and x4 = with x1 < x2 < x3 < x4. question help: worked example 1 submit question jump to answer

Answer

Explanation:

Step1: Apply double - angle formula

Recall that $\sin2x = 2\sin x\cos x$. The given equation $\sin2x+\cos x = 0$ becomes $2\sin x\cos x+\cos x=0$.

Step2: Factor out $\cos x$

Factor the left - hand side: $\cos x(2\sin x + 1)=0$.

Step3: Set each factor equal to zero

Case 1: $\cos x=0$. In the interval $[0,2\pi)$, when $\cos x = 0$, $x=\frac{\pi}{2}$ or $x=\frac{3\pi}{2}$. Case 2: $2\sin x+1 = 0$, so $\sin x=-\frac{1}{2}$. In the interval $[0,2\pi)$, when $\sin x=-\frac{1}{2}$, $x=\frac{7\pi}{6}$ or $x=\frac{11\pi}{6}$.

Answer:

$x_1=\frac{\pi}{2}$ $x_2=\frac{7\pi}{6}$ $x_3=\frac{3\pi}{2}$ $x_4=\frac{11\pi}{6}$