The angle between the curves `y=sin x and y = cos x, 0 lt x lt (pi)/(2)`, is

Options

A. ` pm tan^(-1) sqrt(2) `
B. ` pm tan^(-1) 2sqrt(2) `
C. ` pm "tan"^(-1) (1)/(sqrt(2)) `
D. none of these

Answer

B

Explanation

Clearly, curves `y=sinx " and " y=cosx ` intersect at ` x=(pi)/(4) ` in ` [0,pi].` Thus, the coordinates of the point P of intersection of the two curves are ` P(pi//4, 1//sqrt(2))`
Clearly,
` m_(1)=((dy)/(dx))_(P) = "cos" (pi)/(4) = (1)/ (sqrt(2)) ` for the curve ` y= sinx. `
and,
` m_(2)=((dy)/(dx))_(P) =- "sin"(pi)/(4)= -(1)/(sqrt(2)) ` for the curves ` y=cosx . `
Let ` theta ` be the angle of intersection of the two curves at point P.
Then,
` tan theta = |(m_(1)-m_(2))/(1+m_(1)m_(2))| `
` rArr tan theta = |((1)/(sqrt(2))+(1)/(sqrt(2)))/(1- (1)/(2))|=2 sqrt(2) `
` rArr theta = tan^(-1)(pm 2sqrt(2))=pm tan^(-1)(2sqrt(2)) `