Write the Value of \[\Cos^{- 1} \Left( \Frac{1}{2} \Right) + 2 \Sin^{- 1} \Left( \Frac{1}{2} \Right)\]. - Mathematics

Question

Write the value of

$\cos^{- 1} (\frac{1}{2}) + 2 \sin^{- 1} (\frac{1}{2})$ .

Solution

We have

$\cos^{- 1} \frac{1}{2} + 2 \sin^{- 1} \frac{1}{2}$
$= \cos^{- 1} (\cos \frac{π}{3}) + 2 \sin^{- 1} (\sin \frac{π}{6})$

$[∵ The range of sine is [- \frac{π}{2}, \frac{π}{2}]; \frac{π}{6} \in [- \frac{π}{2}, \frac{π}{2}] and the range of cosine is [0, π]; \frac{π}{3} \in [0, π]]$
$= \frac{π}{3} + 2 (\frac{π}{6})$
$= \frac{π}{3} + \frac{π}{3}$
$= \frac{2 π}{3}$

∴ $\cos^{- 1} (\frac{1}{2}) + 2 \sin^{- 1} (\frac{1}{2}) = \frac{2 π}{3}$

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Inverse Trigonometric Functions (Simplification and Examples)

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Q 11 Q 10 Q 12

Chapter 4: Inverse Trigonometric Functions - Exercise 4.15 [Page 117]

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RD Sharma Mathematics [English] Class 12

Chapter 4 Inverse Trigonometric Functions
Exercise 4.15 | Q 11 | Page 117

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