Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>Let <em>a</em> = sin α, <em>b</em> = cos α and <em>c</em> = √(1 + sin α cos α). In a triangle with sides <em>a</em>, <em>b</em>, <em>c</em>, what is the angle <em>C</em>?</p>
<p>60°</p>
<p>90°</p>
<p>120°</p>
<p>150°</p>
Step-by-Step Solution
Key Concept: Use the law of cosines: c² = a² + b² - 2ab·cos(C). Since a² + b² = sin²α + cos²α = 1 and c² = 1 + sin α cos α, you can directly solve for cos(C) and identify the angle.
<p><strong>Step 1:</strong> Identify the sides: a = sin α, b = cos α, c = √(1 + sin α cos α)</p><p><strong>Step 2:</strong> Apply law of cosines: c² = a² + b² - 2ab·cos(C)</p><p><strong>Step 3:</strong> Calculate each term:</p><p>• c² = 1 + sin α cos α</p><p>• a² + b² = sin²α + cos²α = 1</p><p>• 2ab = 2·sin α·cos α = sin 2α</p><p><strong>Step 4:</strong> Substitute into law of cosines:</p><p>1 + sin α cos α = 1 - 2sin α cos α·cos(C)</p><p>sin α cos α = -2sin α cos α·cos(C)</p><p><strong>Step 5:</strong> For sin α cos α ≠ 0, divide both sides:</p><p>1 = -2cos(C)</p><p>cos(C) = -1/2</p><p><strong>Step 6:</strong> Since C is an angle in a triangle, C ∈ (0, π)</p><p>∴ C = 2π/3 or 120°</p><p><strong>Answer: C</strong></p>
Correct Answer: C