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Question

Mathematics Question on Inverse Trigonometric Functions

Find the principal value of cos-1(−12)\bigg(-\frac{1}{2}\bigg)

Answer

Let cos-1(−12)=y\bigg(-\frac{1}{2}\bigg)=y
Then cos y=−12=−cos(π3)=cos(π−π3)=cos(2π3)cos \,y = -\frac{1}{2}=-cos\bigg(\frac {\pi}{3}\bigg)=cos\bigg(\pi-\frac{\pi}{3}\bigg)=cos\bigg(\frac{2\pi}{3}\bigg)
We know that the range of the principal value branch of
cos-1 is [0,π]and cos(2π3)=−12.\bigg[0,\pi\bigg] and \, cos \bigg(\frac{2\pi}{3}\bigg)=-\frac{1}{2}.
Therefore, the principal value of cos-1 (−12)\bigg(-\frac{1}{2}\bigg) is 2π3\frac {2\pi}{3}