find the exact value: $sin^{-1}\frac{sqrt{2}}{2}$\napproximate $cos^{-1}0.31$ to three decimals (in radians)

find the exact value: $sin^{-1}\frac{sqrt{2}}{2}$\napproximate $cos^{-1}0.31$ to three decimals (in radians)

find the exact value: $sin^{-1}\frac{sqrt{2}}{2}$\napproximate $cos^{-1}0.31$ to three decimals (in radians)

Answer

Explanation:

Step1: Recall the range of inverse sine function

The range of (y = \sin^{-1}(x)) is (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]). We know that (\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}). Since (\frac{\pi}{4}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right])

Step2: Use the definition of inverse - sine function

By the definition of the inverse - sine function (y=\sin^{-1}(x)) means (\sin(y) = x) where (y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]). For (x = \frac{\sqrt{2}}{2}), we have (\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)=\frac{\pi}{4})

Step3: Use a calculator for (\cos^{-1}(0.31))

Set the calculator to radian mode. Then, using the inverse - cosine function on the calculator, (\cos^{-1}(0.31)\approx1.266) (rounded to three decimal places)

Answer:

  1. (\frac{\pi}{4})
  2. (1.266)