# How do you use the sum and difference identities to find the exact value of cos 75?

Dec 6, 2014

The sum and difference identities are given as follows:

sum:
$\cos \left(\alpha + \beta\right) = \cos \left(\alpha\right) \cos \left(\beta\right) - \sin \left(\alpha\right) \sin \left(\beta\right)$

difference:
$\cos \left(\alpha - \beta\right) = \cos \left(\alpha\right) \cos \left(\beta\right) + \sin \left(\alpha\right) \sin \left(\beta\right)$

If you have an angle whose trig values you don't know, but that can be made by two angles you do know, you will use one of these identities. ${75}^{o}$ can be made of the sum of ${45}^{o}$ and ${30}^{o}$, both of which are on the unit circle!

$\cos \left({75}^{o}\right) = \cos \left({45}^{o} + {30}^{o}\right)$

Using our sum identity, where $\alpha = {45}^{o}$ and $\beta = {30}^{o}$;

$= \cos \left({45}^{o}\right) \cos \left({30}^{o}\right) - \sin \left({45}^{o}\right) \sin \left({30}^{o}\right)$

Now just find these values on your unit circle and you get;

$= \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2}$

and the rest is arithmetic.

To use the difference identity, I would recommend using the angles;

$\cos \left({75}^{o}\right) = \cos \left({120}^{o} - {45}^{o}\right)$

Then, using difference identity;

$\cos \left({120}^{o}\right) \cos \left({45}^{o}\right) + \sin \left({120}^{o}\right) \sin \left({45}^{o}\right)$

Which gives;

$- \frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}$

Both expressions simplify to;

$\frac{\sqrt{2} \left(\sqrt{3} - 1\right)}{4}$