evaluate the following expressions. your answer must be an angle -π/2 ≤ θ ≤ π in radians, written as a…

evaluate the following expressions. your answer must be an angle -π/2 ≤ θ ≤ π in radians, written as a multiple of π. note that π is already provided in the answer so you simply have to fill in the appropriate multiple. e.g. if the answer is π/2 you should enter 1/2. do not use decimal answers. write the answer as a fraction or integer. sin⁻¹(sin(7π/6)) = sin⁻¹(sin(5π/3)) = cos⁻¹(cos(3π/4)) = cos⁻¹(cos(2π/3)) =

evaluate the following expressions. your answer must be an angle -π/2 ≤ θ ≤ π in radians, written as a multiple of π. note that π is already provided in the answer so you simply have to fill in the appropriate multiple. e.g. if the answer is π/2 you should enter 1/2. do not use decimal answers. write the answer as a fraction or integer. sin⁻¹(sin(7π/6)) = sin⁻¹(sin(5π/3)) = cos⁻¹(cos(3π/4)) = cos⁻¹(cos(2π/3)) =

Answer

Explanation:

Step1: Recall inverse - sine property

The property of the inverse - sine function (y = \sin^{-1}(\sin(x))) is that (\sin^{-1}(\sin(x))=x) when (-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}). First, find an equivalent angle of (\frac{7\pi}{6}) in the range ([-\frac{\pi}{2},\frac{\pi}{2}]). (\sin(\frac{7\pi}{6})=\sin(\pi + \frac{\pi}{6})=-\sin(\frac{\pi}{6})). And (\sin^{-1}(\sin(\frac{7\pi}{6}))=\sin^{-1}(-\frac{1}{2})). Since (\sin^{-1}(y)) has a range of ([-\frac{\pi}{2},\frac{\pi}{2}]) and (\sin(-\frac{\pi}{6})=-\frac{1}{2}), then (\sin^{-1}(\sin(\frac{7\pi}{6}))=-\frac{\pi}{6}).

Step2: Recall inverse - sine property for (\frac{5\pi}{3})

Find an equivalent angle of (\frac{5\pi}{3}) in the range ([-\frac{\pi}{2},\frac{\pi}{2}]). (\frac{5\pi}{3}=2\pi-\frac{\pi}{3}), and (\sin(\frac{5\pi}{3})=\sin(2\pi - \frac{\pi}{3})=-\sin(\frac{\pi}{3})). Since (\sin^{-1}(\sin(x)) = x) for (x\in[-\frac{\pi}{2},\frac{\pi}{2}]), (\sin^{-1}(\sin(\frac{5\pi}{3}))=-\frac{\pi}{3}).

Step3: Recall inverse - cosine property

The property of the inverse - cosine function (y=\cos^{-1}(\cos(x))) is that (\cos^{-1}(\cos(x)) = x) when (0\leq x\leq\pi). For (x = \frac{3\pi}{4}), since (0\leq\frac{3\pi}{4}\leq\pi), (\cos^{-1}(\cos(\frac{3\pi}{4}))=\frac{3\pi}{4}).

Step4: Recall inverse - cosine property for (\frac{2\pi}{3})

Since (0\leq\frac{2\pi}{3}\leq\pi), (\cos^{-1}(\cos(\frac{2\pi}{3}))=\frac{2\pi}{3}).

Answer:

(\sin^{-1}(\sin(\frac{7\pi}{6}))=-\frac{1}{6}\pi) (\sin^{-1}(\sin(\frac{5\pi}{3}))=-\frac{1}{3}\pi) (\cos^{-1}(\cos(\frac{3\pi}{4}))=\frac{3}{4}\pi) (\cos^{-1}(\cos(\frac{2\pi}{3}))=\frac{2}{3}\pi)