The number of solutions of the equation x = ^2x in the interval (0,10) is ___________ Answ…

Question

The number of solutions of the equation $\sin x = \cos^2x$ in the interval $(0,10)$ is \_\_\_\_\_\_\_\_\_\_\_

Answer: 4

Accepted Answer

$$\sin x = 1-\sin^2x \implies \sin^2x+\sin x-1=0 \implies \sin x = \frac{-1\pm\sqrt5}2$$

Since $\sin x\in(-1,1)$, only $\sin x = \dfrac{\sqrt5-1}2 \approx 0.618$ is valid.

Since $3\pi\approx9.42<10$ and $\dfrac{7\pi}2\approx10.99>10$, the interval $(0,10)$ contains $3$ full "humps" of $\sin x$ (i.e. extends a bit past $3\pi$ into the fourth). The horizontal line $y=0.618$ intersects the graph of $\sin x$ 4 times within $(0,10)$.

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