Trigonometry & Inverse Trigonometry
Range of Trigonometric Functions
Grade 11
Question:
<p>If <span style="text-decoration: overline;">3|sinx| - 2|secy| = a - 5cosz</span>, where x, y, z ∈ ℝ and y ≠ (2n + 1)π/2, n ∈ I, then possible value(s) of a can be</p>
Step-by-Step Solution
Key Concept: Find the range of the left side expression using trigonometric bounds, then match with the range of the right side to determine a.
<p>Analyze the ranges: <span style="text-decoration: overline;">|sinx| ∈ [0,1], so 3|sinx| ∈ [0,3]</span></p><p><span style="text-decoration: overline;">|secy| ≥ 1, so 2|secy| ≥ 2</span></p><p><span style="text-decoration: overline;">cosz ∈ [-1,1], so 5cosz ∈ [-5,5]</span></p><p>Left side: <span style="text-decoration: overline;">3|sinx| - 2|secy| ≤ 3 - 2 = 1</span></p><p>Right side maximum when cosz = -1: a + 5</p><p>Right side minimum when cosz = 1: a - 5</p><p>Therefore a = 6. Answer is (S) 6</p>
Correct Answer: S