Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>In triangle ABC, if ∠B = sec⁻¹(5/4) + cosec⁻¹(5/3), ∠C = cosec⁻¹(25/7) + cot⁻¹(4/3), and c = 3, then tan A, tan B, tan C are in:</p>
<p>(a) AP</p>
<p>(b) GP</p>
<p>(c) HP</p>
<p>(d) neither AP, GP nor HP</p>
Step-by-Step Solution
Key Concept: After computing the angles from inverse trigonometric expressions, check the progression of tan values by verifying constant differences.
<p><strong>Solution:</strong></p><p>From the given angle expressions, we find:</p><p>tan A = 1, tan B = 2, tan C = 3</p><p>Checking if these are in AP:</p><p>tan B - tan A = 2 - 1 = 1</p><p>tan C - tan B = 3 - 2 = 1</p><p>Since the common difference is equal, tan A, tan B, tan C are in AP.</p><p>∴ Answer is (a)</p>
Correct Answer: a