Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade None
Question:
<p>The value of \(\sin^{-1}\left(\dfrac{12}{13}\right) - \sin^{-1}\left(\dfrac{3}{5}\right)\) is equal to:</p>
<p>\(\pi - \sin^{-1}\left(\dfrac{63}{65}\right)\)</p>
<p>\(\dfrac{\pi}{2} - \sin^{-1}\left(\dfrac{56}{65}\right)\)</p>
<p>\(\dfrac{\pi}{2} - \cos^{-1}\left(\dfrac{9}{65}\right)\)</p>
<p>\(\pi - \cos^{-1}\left(\dfrac{33}{65}\right)\)</p>
Step-by-Step Solution
Key Concept: Use the sine difference formula: sin(A - B) = sin A cos B - cos A sin B, where A = sin⁻¹(12/13) and B = sin⁻¹(3/5). Find the complementary cosine values using the Pythagorean identity.
<p><strong>Step 1:</strong> Let A = sin⁻¹(12/13) and B = sin⁻¹(3/5).</p><p>Then sin A = 12/13 and sin B = 3/5.</p><p><strong>Step 2:</strong> Find cos A and cos B using cos²θ + sin²θ = 1.</p><p>cos A = √(1 - 144/169) = √(25/169) = 5/13 (positive since A ∈ [0, π/2])</p><p>cos B = √(1 - 9/25) = √(16/25) = 4/5 (positive since B ∈ [0, π/2])</p><p><strong>Step 3:</strong> Apply sin(A - B) = sin A cos B - cos A sin B.</p><p>sin(A - B) = (12/13)(4/5) - (5/13)(3/5)</p><p>= 48/65 - 15/65</p><p>= 33/65</p><p><strong>Step 4:</strong> Therefore, sin⁻¹(12/13) - sin⁻¹(3/5) = sin⁻¹(33/65)</p><p>∴ Answer: B</p>
Correct Answer: B