Combining two trig terms into one?

[DWill]

If I have 2 cosine terms added together, how would I combine them into one cosine term?

Ex:
A) 3 cos(2t)
B) cos(2t - pi/2)

Thanks

PS. I don't think the sum to product formulas work, I'm wondering how to combine them into a single cosine term?

 

[Gerenuk]

These kind of operations are easiest with complex numbers.
$$3\cos(2t)+\cos(2t-\frac{\pi}{2})=\Re\left(3\exp(2ti)+\exp(2ti-\pi i/2)\right)=\Re\left(\exp(2ti)(3-i)\right)$$
$$=\Re\left(\exp(2ti)\sqrt{10}\exp(-i\arctan\frac{1}{3})\right)=\sqrt{10}}\cos(2t-\arctan(\frac{1}{3}))$$

 

[Mentallic]

You can solve this by the use of the auxillary angle technique.

$cos\left((2t)-\pi/2\right)=sin(2t)$ (you can confirm this by expanding the LHS, but this is a trigo identity you may remember having learnt).

Now let $3cos(2t)+sin(2t)\equiv Rsin(2t-\theta)$

expand the RHS and then equate like terms. Solve the system of 2 equations in R and $\theta$ and then you'll have the original equation in terms of just one trigonometrical expression.