Solve the following trigonometric equations for all solutions. Enter exact answers whenever possible. Choose the smallest positive angle for your base angle. (a) csc($\theta$) = $-\frac{2}{\sqrt{2}}$ Solution with smaller positive base angle: $\theta$ = $\frac{5\pi}{4}$ + 2$\pi$n for all integers n. Solution with larger positive base angle: $\theta$ = $\frac{7\pi}{4}$ + 2$\pi$n for all integers n. (b) sec($\theta$) = $-\frac{2}{\sqrt{3}}$ Solution with smaller positive base angle: $\theta$ = + 2$\pi$n for all integers n. Solution with larger positive base angle: $\theta$ = + 2$\pi$n for all integers n. (c) 5 - 2sin($\theta$) = $\sqrt{2}$ + 5 Solution with smaller positive base angle: $\theta$ = + 2$\pi$n for all integers n. Solution with larger positive base angle: $\theta$ = + 2$\pi$n for all integers n. (d) 5 + tan($\theta$) = $\sqrt{3}$ + 5 Solution with smallest positive base angle: $\theta$ = + $\pi$n for all integers n. (e) -5 + sec($\theta$) = -5 - $\frac{2\sqrt{3}}{3}$ Solution with smaller positive base angle: $\theta$ = + 2$\pi$n for all integers n. Solution with larger positive base angle: $\theta$ = + 2$\pi$n for all integers n.
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To find the solutions, we need to determine the angles where the sine function equals \( \frac{1}{2} \). This occurs at angles where the reference angle is \( \frac{\pi}{6} \) or \( 30^\circ \). The solutions are: \( x = \frac{\pi}{6} + 2\pi n \) for all integers Show more…
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