JEE Main 2022 (Online) 29th June Evening ShiftQuestion 1
The number of solutions of the equation $\sin x=\cos^{2}x$ in the interval $(0,10)$ is $\_\_\_\_\_\_$ .
Trigonometric Ratio & Identities
Answer$4$
Solution
Concept. Solve for $\sin x$ and list the solutions below $10$. Formulas used. (1) $\cos^2x=1-\sin^2x$Step 1 - Solve. By (1), $\sin^2x+\sin x-1=0$, so $\sin x=\frac{\sqrt5-1}{2}\approx0.618$ (the other root is below $-1$). Step 2 - List. With $\alpha=\sin^{-1}0.618\approx0.666$: $$\alpha\approx0.666,\ \pi-\alpha\approx2.475,\ 2\pi+\alpha\approx6.949,\ 3\pi-\alpha\approx8.758.$$ The next one, $4\pi+\alpha\approx13.2$, is beyond $10$. Step 3 - Count. $4$. Answer: 4