Aspire Faculty ID #18302 · Topic: AMU MCA 2017 · Just now
AMU MCA 2017

Using the information from the figure, $\cos^{-1} x$ is equal to

Solution

From the figure,
$ BC=\sqrt{AB^2-AC^2}=\sqrt{1-x^2} $

$ \Rightarrow \cos\alpha=\sqrt{1-x^2},\ \sin\alpha=x $

$ \Rightarrow \cos\beta=x,\ \sin\beta=\sqrt{1-x^2} $

$ \Rightarrow \sin^{-1}x=\alpha,\ \cos^{-1}x=\beta $

In right triangle,

$ \alpha+\beta=\frac{\pi}{2} $

$ \Rightarrow \sin^{-1}x+\cos^{-1}x=\frac{\pi}{2} $

$ \Rightarrow \cos^{-1}x=\frac{\pi}{2}-\sin^{-1}x $
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