Question 1591
CBSE Class 11 Mathematics · Trigonometric Functions · Multiple and half angles
Complete every requested part of miscellaneous exercise, question 9: cos x = − , x in quadrant III 3 3 1
Solution to Question 1591
Complete workingProceed from the given expression and keep every transformation explicit.
\[\cos x=-\frac{1}{3},\ x\in Q_{III}\Rightarrow \frac{x}{2}\in Q_{II}\]
\[\begin{aligned}\sin(\frac{x}{2}) &= \sqrt{(1-\cos x)/2}\\&= \sqrt{\frac{2}{3}}\end{aligned}\]
\[\begin{aligned}\cos(\frac{x}{2}) &= -\sqrt{(1+\cos x)/2}\\&= -1/\sqrt3\end{aligned}\]
\[\tan(\frac{x}{2})=-\sqrt2\]
The last line satisfies the required domain and establishes the result.
Final answer
\[\sin(\frac{x}{2})=\sqrt{\frac{2}{3}},\cos(\frac{x}{2})=-1/\sqrt3,\tan(\frac{x}{2})=-\sqrt2\]