Sketch r = cos(theta) + sin(2theta) with proper justification?

I would appreciate if there is something like a table of values and reasoning behind how you graph it

1 Answer
Jun 17, 2018

Please see below.

Explanation:

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Since we do not know what it looks lie, let's start with finding its key points. We can find r for values of theta equal to special angles and their multiples within the domain 0 < theta < 2pi:

r=costheta+sin2theta

theta=0, :. r=cos(0)+sin(0)=1+0=1

theta=pi/6, :. r=cos(pi/6)+sin(pi/3)=sqrt3/2+sqrt3/2=sqrt3=1.73

theta=pi/4, :. r=sqrt2/2+1=1.71

theta=pi/3, :. r=1/2+sqrt3/2=1.37

theta=pi/2, :. r=0+0=0

theta=(2pi)/3, :. r=-1/2-sqrt3/2=-1.37

theta=(3pi)/4, :. r=-sqrt2/2-1=-1.71

theta=(5pi)/6, :. r=-sqrt3/2-sqrt3/2=-sqrt3=-1.73

theta=pi, :. r=-1+0=-1

Similarly, you can continue finding points on the grid by trying more values of theta until 2pi.

Now, let's see what values of theta gives us r=0:

costheta+sin2theta=0

costheta+2sinthetacostheta=0

costheta(1+2sintheta)=0

costheta=0, :. theta=pi/2, (3pi)/2

1+2sintheta=0, :. sintheta=-1/2, :. theta=(7pi)/6, (11pi)/6

This means the graph will pass through the origin four times between 0 < theta < 2pi

Now, let's see what the maximum and minimum values of r are by taking the derivative of the function, setting it equal to 0, and finding the roots:

(dr)/(d theta)=-sintheta+2cos2theta=0

2cos2theta=sintheta

2(1-2sin^2theta)=sintheta

4sin^2theta+sintheta-2=0

Using the quadratic formula:

sintheta=0.59307 and -0.84307

theta=0.635 and -1.003 radians

theta~~pi/5 and -pi are where the maximum and minimum r will occur which are r~~1.76 and 0

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