Trigonometry & Inverse Trigonometry
Properties of Inverse Trigonometric Functions
Grade 12
Question:
<p>The value of \( \tan^{-1}\left(\dfrac{1}{4}\right) + \tan^{-1}\left(\dfrac{2}{9}\right) \) is:</p>
<p>(A) \(\tan^{-1}\left(\dfrac{1}{3}\right)\)</p>
<p>(B) \(\tan^{-1}\left(\dfrac{1}{2}\right)\)</p>
<p>(C) \(\tan^{-1}\left(\dfrac{17}{34}\right)\)</p>
<p>(D) \(\dfrac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Use the addition formula for inverse tangent: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)) when ab < 1, which simplifies the sum of two inverse tangent functions into a single term.
<p><strong>Step 1:</strong> Identify the addition formula for inverse tangent. For tan⁻¹(a) + tan⁻¹(b), use: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)) when ab < 1</p><p><strong>Step 2:</strong> Check the condition: a = 1/4, b = 2/9, so ab = (1/4)(2/9) = 2/36 = 1/18 < 1 ✓</p><p><strong>Step 3:</strong> Calculate the numerator: a + b = 1/4 + 2/9 = 9/36 + 8/36 = 17/36</p><p><strong>Step 4:</strong> Calculate the denominator: 1 - ab = 1 - 1/18 = 17/18</p><p><strong>Step 5:</strong> Apply the formula: tan⁻¹((17/36)/(17/18)) = tan⁻¹((17/36) × (18/17)) = tan⁻¹(18/36) = tan⁻¹(1/2)</p><p>∴ Answer: B</p>
Correct Answer: B