   Chapter 12.3, Problem 32E

Chapter
Section
Textbook Problem

# Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.)32. y = sin x, y cos x, 0≤ x ≤ π/2

To determine

To find: The acute angle between the curves and point of intersection.

Explanation

Given:

y=sinx (1)

y=cosx (2)

Formula:

Write the expression to find ab in terms of θ .

ab=|a||b|cosθ

Here,

|a| is the magnitude of a vector,

|b| is the magnitude of b vector, and

θ is the angle between vectors a and b.

Rearrange equation.

θ=cos1(ab|a||b|) (3)

Consider a general expression to find dot product between two two-dimensional vectors.

ab=a1,a2b1,b2

ab=a1b1+a2b2 (4)

Consider a general expression to find magnitude of a two dimensional vector that is a=a1,a2 .

|a|=a12+a22 (5)

Similarly, Consider a general expression to find magnitude of a two dimensional vector that is b=b1,b2 .

|b|=b12+b22 (6)

Substitute equation (1) in equation (2).

sinx=cosx

Rearrange the equation.

sinxcosx=1tanx=1x=tan1(1)x=π4

At 0xπ2 .

In equation (1), substitute π4 for x .

y=sin(π4)=22

In equation (2), substitute π4 for x .

y=cos(π4)=22

Thus, the point of intersection is (π4,22)_

Differentiate equation (1) with respect to x .

ddx(y)=ddx(sinx)

ddx(y)=cosx

Substitute π4 for x .

ddx(y)=cos(π4)=22

Differentiate equation (2) with respect to x .

ddx(y)=ddx(cosx)

ddx(y)=sinx

Substitute π4 for x

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