Find the smallest positive roots of the equation $\sqrt{\sin(1-x)}=\sqrt{\cos x}$
- $x=\displaystyle \frac{7\pi}{4}-\frac{1}{2}$.
- $x=\displaystyle \frac{7\pi}{4}+\frac{1}{2}$.
- $x=\displaystyle \frac{5\pi}{4}+\frac{1}{2}$.
- $x=\displaystyle \frac{5\pi}{4}-\frac{1}{2}$.
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AI explanation
Squaring both sides of the equation gives sin(1 - x) = cos x. Using the co-function identity, write cos x as sin(pi/2 - x). This results in sin(1 - x) = sin(pi/2 - x), which implies 1 - x = pi/2 - x + 2k pi or 1 - x = pi - (pi/2 - x) + 2k pi. The first case simplifies to 1 = pi/2 + 2k pi, which is impossible for any integer k. Solving the second case yields 1 - x = pi/2 + x + 2k pi, meaning 2x = 1 - pi/2 - 2k pi, so x = 1/2 - pi/4 - k pi. Testing the smallest positive values by substituting k = -1 and k = -2 gives x = 1/2 + 3 pi/4 and x = 1/2 + 7 pi/4, both of which satisfy the original equation. The smallest of these positive roots is x = 7 pi/4 + 1/2.