Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade None
Question:
<p>If \(\alpha = \theta_1 + \theta_2\) and \(x = \theta_1 - \theta_2\) and \(\tan\theta_1 = \lambda\tan\theta_2\), then \(\sin x : \sin\alpha\) is equal to</p>
<p>(a) \(\frac{\lambda-1}{\lambda+1}\)</p>
<p>(b) \(\frac{\lambda+1}{\lambda-1}\)</p>
<p>(c) \(\frac{1}{\lambda}\)</p>
<p>(d) \(\frac{\lambda}{\lambda+1}\)</p>
Step-by-Step Solution
Key Concept: Use the sum-to-product identity combined with the constraint tan θ₁ = λ tan θ₂ to express sin x and sin α in terms of a common parameter, then find their ratio by converting the tangent condition into a relationship between sines and cosines.
<p><strong>Step 1:</strong> Write the given constraint: tan θ₁ = λ tan θ₂, which gives</p><p>sin θ₁/cos θ₁ = λ sin θ₂/cos θ₂</p><p>⟹ sin θ₁ cos θ₂ = λ sin θ₂ cos θ₁ ... (i)</p><p><strong>Step 2:</strong> Express sin x and sin α:</p><p>sin x = sin(θ₁ - θ₂) = sin θ₁ cos θ₂ - cos θ₁ sin θ₂</p><p>sin α = sin(θ₁ + θ₂) = sin θ₁ cos θ₂ + cos θ₁ sin θ₂</p><p><strong>Step 3:</strong> Substitute equation (i) into both expressions:</p><p>sin x = λ sin θ₂ cos θ₁ - cos θ₁ sin θ₂ = cos θ₁ sin θ₂(λ - 1)</p><p>sin α = λ sin θ₂ cos θ₁ + cos θ₁ sin θ₂ = cos θ₁ sin θ₂(λ + 1)</p><p><strong>Step 4:</strong> Find the ratio:</p><p>sin x : sin α = cos θ₁ sin θ₂(λ - 1) : cos θ₁ sin θ₂(λ + 1)</p><p>= (λ - 1) : (λ + 1)</p><p>∴ Answer: A</p>
Correct Answer: A