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Question

Mathematics Question on Inverse Trigonometric Functions

Find the principal value of cos-1(−12)\bigg(-\frac{1}{\sqrt2}\bigg)

Answer

Let cos-1(−12)=y\bigg(-\frac{1}{\sqrt2}\bigg)=y.
Then cos y=−12=−cos(π4)=cos(π−π4)=cos(3π4)cos\, y= -\frac{1}{\sqrt2}=-cos\bigg(\frac{\pi}{4}\bigg)=cos\bigg(\pi-\frac{\pi}{4}\bigg)=cos\bigg(\frac{3\pi}{4}\bigg).
We know that the range of the principal value branch of cos-1 is
[0,π]and cos(3π4)=−12.\bigg[0,\pi\bigg]and\, cos \bigg(\frac{3\pi}{4}\bigg)=-\frac{1}{\sqrt2}.
Therefore, the principal value of cos-1(−12)\bigg(-\frac{1}{\sqrt2}\bigg) is 3π4.\frac{3\pi}{4}.