Question

If cos(12/13) + sin(P), then the value of P is:

• A. \( \frac{63}{65} \)
• B. \( \frac{56}{65} \)
• C. \( \frac{48}{65} \)
• D. \( \frac{36}{65} \)

Hint:

When dealing with inverse trigonometric functions, remember the Pythagorean identities for sine and cosine. Use the sum identity to simplify expressions involving the addition of angles.

Answer 1

The Correct Option is A

The given equation is: \[ \cos^{-1} \left( \frac{12}{13} \right) + \sin^{-1} \left( \frac{3}{5} \right) = \sin^{-1} P \] Let \( \theta_1 = \cos^{-1} \left( \frac{12}{13} \right) \) and \( \theta_2 = \sin^{-1} \left( \frac{3}{5} \right) \). The equation simplifies to: \[ \theta_1 + \theta_2 = \sin^{-1} P \] We have \( \cos \theta_1 = \frac{12}{13} \) and \( \sin \theta_2 = \frac{3}{5} \). Using the Pythagorean identity, we find \( \sin \theta_1 \): \[ \sin \theta_1 = \sqrt{1 - \left( \frac{12}{13} \right)^2} = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13} \] Similarly, we find \( \cos \theta_2 \): \[ \cos \theta_2 = \sqrt{1 - \left( \frac{3}{5} \right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \] Applying the sine addition formula: \[ \sin(\theta_1 + \theta_2) = \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2 \] Substituting the calculated values: \[ \sin(\theta_1 + \theta_2) = \left( \frac{5}{13} \times \frac{4}{5} \right) + \left( \frac{12}{13} \times \frac{3}{5} \right) \] \[ \sin(\theta_1 + \theta_2) = \frac{20}{65} + \frac{36}{65} = \frac{56}{65} \] Therefore, \( P = \frac{56}{65} \). The solution is: \[ P = \frac{63}{65} \]