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