Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <i>2x</i> and <i>2y</i> are complementary angles and tan(<i>x</i> + 2<i>y</i>) = 2, then which of the following can be correct?</p>
<p>(a) cos(<i>x</i> - <i>y</i>) = 1/√2</p>
<p>(b) tan(<i>x</i> - <i>y</i>) = 1/3</p>
<p>(c) sin(<i>x</i> - <i>y</i>) = -1/√2</p>
<p>(d) cot(<i>x</i> - <i>y</i>) = 2/3</p>
Step-by-Step Solution
Key Concept: Use the complementary angle condition to establish a relationship between x and y, then apply the given tangent condition to solve for individual angles.
<p><strong>Step 1:</strong> Given 2x + 2y = π/2 (complementary angles), so x + y = π/4</p><p><strong>Step 2:</strong> Given tan(x + 2y) = 2. Since x + y = π/4, we have x + 2y = π/4 + y</p><p><strong>Step 3:</strong> Therefore tan(π/4 + y) = 2</p><p>⟹ (1 + tan y)/(1 - tan y) = 2</p><p>⟹ 1 + tan y = 2(1 - tan y)</p><p>⟹ 3 tan y = 1, so tan y = 1/3</p><p><strong>Step 4:</strong> Since x + y = π/4, we have x = π/4 - y, so x - y = π/4 - 2y</p><p><strong>Step 5:</strong> Using tan(x - y) = (tan x - tan y)/(1 + tan x tan y) with the relationship tan x = (1 - tan y)/(1 + tan y):</p><p>tan(x - y) = 1/3</p><p>∴ Answer is B.</p>
Correct Answer: B