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Inverse Trigonometric Functions Example 05

Inverse Trigonometric Functions Example 05

Find the exact value of \theta in the equation 2\cos^{-1}(\theta) + \sin^{-1}(\theta) = \frac{\pi}{3}.


Step 1: \quad  2\cos^{-1}(\theta) = \frac{\pi}{3} - \sin^{-1}(\theta)\\
Step 2: \quad  \cos^{-1}(\theta) = \frac{\frac{\pi}{3} - \sin^{-1}(\theta)}{2} \\
Step 3: \quad  \cos^{-1}(\theta) = \frac{\pi}{6} - \sin^{-1}(\theta)\\
Step 4: \quad  \cos^{-1}(\theta) = \frac{\pi}{6} - \theta\\
Step 5: \quad  \theta = \frac{\sqrt{3}}{2}\cos(\theta) + \frac{1}{2}\sin(\theta)

\textbf{Note:} Solving the equation in Step 5 involves a transcendental equation, which may require numerical methods for an approximate solution.

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