The number of roots of the equation (81)^^2x+(81)^^2x=30 in the interval [0,] is equal to:

Question

The number of roots of the equation $(81)^{\sin^2x}+(81)^{\cos^2x}=30$ in the interval $[0,\pi]$ is equal to:

Accepted Answer

Let $y=81^{\sin^2x}$, so $81^{\cos^2x} = \dfrac{81}y$. Then $y+\dfrac{81}y=30 \implies y^2-30y+81=0 \implies (y-27)(y-3)=0$.

Case $y=3$: $81^{\sin^2x}=3 \implies 3^{4\sin^2x}=3 \implies \sin^2x=	frac14 \implies x=	frac\pi6,	frac{5\pi}6$

Case $y=27$: $3^{4\sin^2x}=3^3 \implies \sin^2x=	frac34 \implies x=	frac\pi3,	frac{2\pi}3$

Total: 4 solutions.

Solutions · 1