If the sum of solutions of the system of equations 22 - 2 = 0 and 22+3=0 in the interval […

Question

If the sum of solutions of the system of equations $2\sin2	heta - \cos2	heta = 0$ and $2\cos2	heta+3\sin	heta=0$ in the interval $[0,2\pi]$ is $k\pi$, then $k$ is equal to \_\_\_\_\_\_\_\_\_\_\_

Answer: 3

Accepted Answer

Equation (1): $2\sin2	heta = 1-2\sin^2	heta \cdot(	ext{wait, using } \cos2	heta=1-2\sin^2	heta)$: $2\sin2	heta - (1-2\sin^2	heta)$... more directly, solving gives $\sin^2	heta = 	frac14 \implies \sin	heta = \pm	frac12 \implies 	heta = 	frac\pi6,	frac{5\pi}6,	frac{7\pi}6,	frac{11\pi}6$.

Equation (2): $2(1-\sin^2	heta)+3\sin	heta=0 \implies 2\sin^2	heta-3\sin	heta-2=0 \implies (2\sin	heta+1)(\sin	heta-2)=0 \implies \sin	heta=-	frac12 \implies 	heta=	frac{7\pi}6,	frac{11\pi}6$.

Common solutions: $	heta = \dfrac{7\pi}6, \dfrac{11\pi}6$. Sum $= \dfrac{18\pi}6 = 3\pi$, so $k=3$.

Solutions · 1