How to draw complicated functions by hand?

Question

I'm to visualize what

f(t) = 2+cos(3t-6/pi) + 1/4cos(t/2+pi/3)+sin^2(t)

looks like, but I'm not sure how it's done by hand. I gathered that the function is periodic at 4pi and that the periods of the sin and cos -functions are

T(cos(3t-6/pi)) = 2/3 * pi
T(1/4cos(t/2+pi/3)) = 4 * pi
T(sin(t)) = 2 * pi

Is the only way to do this by hand to draw each function here, evaluate how they act when they are added together and account for the +2 outside of the function?

Answer 1

To have an idea of what the function looks like:

• find the periodicity of the function (you seem to have already done that)

• differentiate the function

• find the sign of $f'$

• you will then know the variations of the function.

It will look something like this: