Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>Given expression = \(1 + 2^2 + 1 + 3^2 + \text{cosec}\left(\tan^{-1}\dfrac{4}{3} + \tan^{-1}\dfrac{4}{3}\right)\). Find the value of the expression.</p>
<p>\(15 + \dfrac{24}{25}\)</p>
<p>\(15 + \dfrac{25}{24}\)</p>
<p>\(16 + \dfrac{25}{24}\)</p>
<p>\(15 + \dfrac{25}{24}\)</p>
Step-by-Step Solution
Key Concept: Convert inverse trigonometric functions using the tangent addition formula: tan(A+B) = (tanA + tanB)/(1 - tanA·tanB), then find cosecant of the resulting angle using a right triangle.
<p><strong>Step 1:</strong> Simplify the algebraic part: 1 + 2² + 1 + 3² = 1 + 4 + 1 + 9 = 15</p><p><strong>Step 2:</strong> Find tan⁻¹(4/3) + tan⁻¹(4/3) using the addition formula:</p><p>Let α = tan⁻¹(4/3), so tan(2α) = (2·tan(α))/(1 - tan²(α)) = (2·(4/3))/(1 - (4/3)²)</p><p><strong>Step 3:</strong> Calculate: tan(2α) = (8/3)/(1 - 16/9) = (8/3)/(-7/9) = (8/3)·(-9/7) = -24/7</p><p><strong>Step 4:</strong> Since tan(2α) = -24/7, construct a right triangle where the opposite side is 24, adjacent side is 7 (taking absolute value). The hypotenuse = √(24² + 7²) = √(576 + 49) = √625 = 25</p><p><strong>Step 5:</strong> Therefore, cosec(2α) = hypotenuse/opposite = 25/24</p><p><strong>Step 6:</strong> Total expression = 15 + 25/24 = (360 + 25)/24 = 385/24</p><p>∴ Answer: D (385/24 or equivalent form)</p>
Correct Answer: D