Question

The measure of the angle between the curves y=2sin²x and y=cos2x at x=π/6 is.....,Select correct option from the given options.
(a) π/2
(b) π/3
(c) π/4
(d) π/6

Answer 1

answer : option (b) π/3

explanation : first find slope of tangent of curves at x = π/6.

for curve, y = 2sin²x

differentiating both sides,

dy/dx = 4sinx.cosx

at x = π/6 , $m_1=\frac{dy}{dx}|_{(x=\pi/6)}$ = 4sin(π/6)cos(π/6)

= 4 × 1/2 × √3/2 = √3

hence, slope of tangent of curve y = 2sin²x , $m_1=\sqrt{3}$

for curve, y = cos2x

differentiating both sides,

dy/dx = -2sin2x

at x = π/6, $m_2=\frac{dy}{dx}|_{(x=\pi/6)}$ = -2sin2(π/6) = -√3

hence, slope of tangent of curve, y = cos2x, $m_2=-\sqrt{3}$

now, angle between the curves , tanθ= $\left|\frac{m_1-m_2}{1+m_1+m_2}\right|$

= |(√3 + √3)/{1 + √3.(-√3)|

= √3 = tan(π/3)

hence, θ = π/3