find the exact value of \\( \\sin ^ { - 1 } \\left( \\sin \\frac { 9 \\pi } { 7 } \\right) \\).\nwrite your…

find the exact value of \\( \\sin ^ { - 1 } \\left( \\sin \\frac { 9 \\pi } { 7 } \\right) \\).\nwrite your answer in radians in terms of \\( \\pi \\).\nif necessary, click on \undefined.\

find the exact value of \\( \\sin ^ { - 1 } \\left( \\sin \\frac { 9 \\pi } { 7 } \\right) \\).\nwrite your answer in radians in terms of \\( \\pi \\).\nif necessary, click on \undefined.\

Answer

Explanation:

Step1: Use the property of sine function

We know that (\sin(x)=\sin(\pi - x)). So, (\sin\frac{9\pi}{7}=\sin(\pi+\frac{2\pi}{7})). Since (\sin(A + B)=\sin A\cos B+\cos A\sin B), here (A=\pi), (B = \frac{2\pi}{7}), (\sin(\pi+\frac{2\pi}{7})=-\sin\frac{2\pi}{7}).

Step2: Use the property of inverse - sine function

The range of (y = \sin^{-1}(x)) is ([-\frac{\pi}{2},\frac{\pi}{2}]). We want to find (y=\sin^{-1}(\sin\theta)). Let (\theta=\frac{9\pi}{7}). We rewrite (\sin\frac{9\pi}{7}) as (-\sin\frac{2\pi}{7}=\sin(-\frac{2\pi}{7})) (because (\sin(-x)=-\sin x)). Since (-\frac{\pi}{2}\leq-\frac{2\pi}{7}\leq\frac{\pi}{2}), by the property (\sin^{-1}(\sin x)=x) for (x\in[-\frac{\pi}{2},\frac{\pi}{2}])

Answer:

(-\frac{2\pi}{7})