Applications of Derivatives
Angle between curves
Grade 12
Question:
<p>The angle between the tangents to the curves \(y = \sin x\) and \(y = \cos x\) at a point of intersection is</p>
<p>(a) \(\dfrac{\pi}{4}\)</p>
<p>(b) \(\tan^{-1}(2\sqrt{2})\)</p>
<p>(c) \(\tan^{-1}\!\left(\dfrac{1}{2\sqrt{2}}\right)\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Find intersection points of sin x and cos x, calculate slopes of tangents at those points using derivatives, then use the angle formula between two lines: tan θ = |m₁ - m₂|/(1 + m₁m₂).
<p><strong>Step 1:</strong> Find points of intersection of y = sin x and y = cos x.</p><p>At intersection: sin x = cos x ⟹ tan x = 1 ⟹ x = π/4 + nπ</p><p><strong>Step 2:</strong> Find slopes of tangents at intersection point x = π/4.</p><p>For y = sin x: m₁ = dy/dx = cos x = cos(π/4) = 1/√2</p><p>For y = cos x: m₂ = dy/dx = -sin x = -sin(π/4) = -1/√2</p><p><strong>Step 3:</strong> Apply angle formula between two lines.</p><p>tan θ = |m₁ - m₂|/(1 + m₁m₂) = |(1/√2) - (-1/√2)|/(1 + (1/√2)(-1/√2))</p><p>tan θ = |2/√2|/(1 - 1/2) = (√2)/(1/2) = 2√2</p><p><strong>Step 4:</strong> Find θ.</p><p>θ = arctan(2√2) = 90° or π/2 radians</p><p>∴ Answer: <strong>B (90° or π/2)</strong></p>
Correct Answer: B