# Addition of two sinusoids

#### Amathproblem22

##### New member
So I have figured out two equations model a Ferris wheel ride(two different ferris wheels).
F1 = Ferris wheel one
F2= Ferris wheel two

F1, $$\displaystyle h=-12\cos\frac{\pi}{10}t+12.5$$

F2, $$\displaystyle h=-12\cos\frac{\pi}{30}t+15$$

Now from these two equations, I want to know when F2+F1= 27.5m i.e I want to find the times when the combined height of each Ferris wheels seat adds up to 27.5m.
How would I go about doing this?

#### MarkFL

##### Administrator
Staff member
If we equate sum of the two height functions to 27.5, we ultimately obtain:

$$\displaystyle \cos\left(\frac{\pi}{10}t\right)+\cos\left(\frac{\pi}{30}t\right)=0$$

Let's let:

$$\displaystyle \theta=\frac{\pi}{30}t$$

And we may write:

$$\displaystyle \cos(3\theta)+\cos(\theta)=0$$

Applying a triple-angle identity for cosine, we ultimately get:

$$\displaystyle 2\cos^3(\theta)-\cos(\theta)=0$$

Factor:

$$\displaystyle \cos(\theta)(2\cos^2(\theta)-1)=0$$

Can you proceed?

#### Amathproblem22

##### New member
If we equate sum of the two height functions to 27.5, we ultimately obtain:

$$\displaystyle \cos\left(\frac{\pi}{10}t\right)+\cos\left(\frac{\pi}{30}t\right)=0$$

Let's let:

$$\displaystyle \theta=\frac{\pi}{30}t$$

And we may write:

$$\displaystyle \cos(3\theta)+\cos(\theta)=0$$

Applying a triple-angle identity for cosine, we ultimately get:

$$\displaystyle 2\cos^3(\theta)-\cos(\theta)=0$$

Factor:

$$\displaystyle \cos(\theta)(2\cos^2(\theta)-1)=0$$

Can you proceed?
$\cos \left(θ\right)=0\quad \mathrm{or}\quad \:2\cos ^2\left(θ\right)-1=0$

$\cos \left(θ\right)=0: θ=\frac{\pi }{2}+2\pi n,\:θ=\frac{3\pi }{2}+2\pi n$

$2\cos ^2\left(θ\right)-1=0: θ=\arccos \left(\sqrt{\frac{1}{2}}\right)+2\pi n,\:θ=2\pi -\arccos \left(\sqrt{\frac{1}{2}}\right)+2\pi n,\:θ=\arccos \left(-\sqrt{\frac{1}{2}}\right)+2\pi n,\:θ=-\arccos \left(-\sqrt{\frac{1}{2}}\right)+2\pi n$ ?

#### MarkFL

##### Administrator
Staff member
Yes, for:

$$\displaystyle \cos(\theta)=0$$

This implies (where $$k\in\mathbb{Z}$$):

$$\displaystyle \theta=\frac{\pi}{2}+\pi k=\frac{\pi}{2}(2k+1)$$

$$\displaystyle t=\frac{30}{\pi}\theta=15(2k+1)$$

And for:

$$\displaystyle \cos(\theta)=\pm\frac{1}{\sqrt{2}}$$

This implies:

$$\displaystyle \theta=\frac{\pi}{4}+\frac{\pi}{2}k=\frac{\pi}{4}(2k+1)$$

$$\displaystyle t=\frac{30}{\pi}\theta=\frac{15}{2}(2k+1)$$