Question #3d967

1 Answer
Nov 24, 2017

Please see below.

Explanation:

Let's set the two functions equal to each other to solve for the x-coordinate of the intersection point:

cos2x=2sin3x

But we know:

sin(a+b)=sinacosb+cosasinb

We can re-write our equation and use the above formula:

cos2x=2sin(2x+x)

cos2x=2(sin2xcosx+cos2xsinx)

cos2x=2sin2xcosx+2cos2xsinx

But we know:

sin2x=2sinxcosx and cos2x=2cos^2x-1. Let's plug them in:

2cos^2x-1=2*2sinxcosxcosx+2(2cos^2x-1)sinx

2cos^2x-1=4sinxcos^2x+4sinxcos^2x-2sinx

2cos^2x-1=8sinxcos^2x-2sinx

Let's use sin^2x+cos^2x=1 and plug in 1-sin^2x for cos^2x to turn the entire equation in terms of sinx:

2(1-sin^2x)-1=8sinx(1-sin^2x)-2sinx

2-2sin^2x-1=8sinx-8sin^3x-2sinx

1-2sin^2x=6sinx-8sin^3x

8sin^3x-2sin^2x-6sinx+1=0

Here is the link for an online calculator for cubic roots:

http://www.1728.org/cubic.htm

You will get the following three answers for sinx:

Answer 1: sinx=(0.91842444565049) which means

x=arcsin(0.91842444565049) or x=66.7 Degrees

Answer 2: sinx=(-0.8320078223739) which means

x=arcsin(-0.8320078223739) or x=-56.3 Degrees

Answer 3: sinx=(0.16358337672342) which means

x=arcsin(0.16358337672342) or x=9.4 Degrees

Only answers 1 and 3 are acceptable because they fall in the specified interval. As such the two curves intersect at two points whose x-values are 66.7 and 9.4 Degrees

To find the angles between the curves at these points you need to find the slopes of the tangent lines to the curves at these points and then find the angle between the two tangents.

To find the slope of a tangent to a curve you take the derivative of the functions and evaluate it at the x value of the tangent point.

For y=cos2x

dy/dx=-2sin2x

For y=2sin3x

dy/dx=6cos3x

At x=9.4 Degrees slope of tangent to y=cos2x is

m_1=-2sin(2*9.4)=-2sin(18.8)=-2*0.32 or m_1=-0.64

At this point, the slope of tangent to y=2sin3x is

m_2=6cos(3*9.4)=6cos(28.2)=6*0.88 or m_2=5.28

Angle between tangent one and x-axis is arctan(m_1)=arctan(-0.64)=-32.62 Degrees

Angle between tangent two and x-axis is arctan(m_2)=79.27 Degrees

Then the angle between the two curves is 79.27-(-32.62)=111.89 Degrees

At x=66.7 Degrees

m_1=-2sin(2*66.7)=-2sin(133.4)=-2*0.73=-1.46

m_2=6cos(3*66.7)=6cos(200.1)=6*-0.94=-5.64

Angle between tangent one and x-axis is arctan(-1.46)=-55.59 Degrees

Angle between tangent two and x-axis is arctan(-5.64)=-79.94 Degrees

Angle between the two curves is -55.59-(-79.94)=24.35 Degrees