Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade 11
Question:
<p>The value of \(\sqrt{3}\csc 20° - \sec 20°\) is equal to</p>
<p>(a) 2</p>
<p>(b) 4</p>
<p>(c) \(2\frac{\sin 20°}{\sin 40°}\)</p>
<p>(d) \(4\frac{\sin 20°}{\sin 40°}\)</p>
Step-by-Step Solution
Key Concept: Convert the expression into a single trigonometric function by expressing √3csc20° - sec20° as a linear combination of sine and cosine, then use the formula asinθ + bcosθ = √(a² + b²)sin(θ + φ).
<p><strong>Step 1:</strong> Rewrite the expression with common denominator</p><p>√3csc20° - sec20° = √3/sin20° - 1/cos20°</p><p>= (√3cos20° - sin20°)/(sin20°·cos20°)</p><p><strong>Step 2:</strong> Recognize the numerator pattern</p><p>√3cos20° - sin20° = 2(√3/2·cos20° - 1/2·sin20°)</p><p>= 2(sin60°cos20° - cos60°sin20°)</p><p>= 2sin(60° - 20°) = 2sin40°</p><p><strong>Step 3:</strong> Simplify using denominator identity</p><p>sin20°·cos20° = (1/2)sin40°</p><p><strong>Step 4:</strong> Combine results</p><p>(2sin40°)/((1/2)sin40°) = 2sin40°/(sin40°/2) = 4</p><p>∴ Answer: D (which equals 4)</p>
Correct Answer: D