Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>Given the equations with solutions in the interval <span style='color:rgb(192,0,0)'>(0, π/2)</span>:</p><p>(i) <span style='color:rgb(192,0,0)'>sin a = 1/2</span>, so <span style='color:rgb(192,0,0)'>cos a = √3/2</span></p><p>(ii) <span style='color:rgb(192,0,0)'>cos β = 1/3</span>, so <span style='color:rgb(192,0,0)'>sin β = √(8/9)</span></p><p>(iii) <span style='color:rgb(192,0,0)'>sin γ = cos γ = 1/√2</span></p><p>(iv) <span style='color:rgb(192,0,0)'>cos γ = 1, sin γ = 0</span></p><p>Find: <span style='color:rgb(192,0,0)'>cos a + cos β + cos γ</span></p>
<p>(A) <span style='color:rgb(0,0,255)'>√3/2 + 1/3 + 1/√2</span></p>
<p>(B) <span style='color:rgb(0,0,255)'>√3/2 + 1/3 + 1</span></p>
<p>(C) <span style='color:rgb(0,0,255)'>(37√6 + 27√2 + 6)/(67√2)</span></p>
<p>(D) <span style='color:rgb(0,0,255)'>(37√3 + 8)/6</span></p>
Step-by-Step Solution
Key Concept: Extract trigonometric values from the given equations and add them together. Alternative solutions exist for γ, requiring identification of the correct case.
<p><strong>Step 1:</strong> From the given equations, we have extracted the trigonometric values:</p><ul><li>From equation (i): <span style='color:rgb(192,0,0)'>cos a = √3/2</span></li><li>From equation (ii): <span style='color:rgb(192,0,0)'>cos β = 1/3</span></li><li>From equations (iii) and (iv): <span style='color:rgb(192,0,0)'>cos γ = 1</span> (as cos γ = 1/√2 and cos γ = 1 are given as alternatives)</li></ul><p><strong>Step 2:</strong> Adding these values:</p><p><span style='color:rgb(192,0,0)'>cos a + cos β + cos γ = √3/2 + 1/3 + 1</span></p><p>∴ Answer is <span style='color:rgb(192,0,0)'>√3/2 + 1/3 + 1</span></p>
Correct Answer: B