# How to calculate $\cos(\pi/4)$ and $\sin(\pi/4)$? [closed]

I need to calculate $\sin(\frac{\pi}{4})$ and $\cos(\frac{\pi}{4})$

only by using:

1) $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$

2) $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$

3) $\cos(a)^2+\sin(a)^2=1$

I am stuck and don't know how to start. I know what the values are, but I have no clue how to calculate them using the three identities. Any hints will be appreciated.

• Do you have any givens? For example, can you use that $sin(\frac{\pi}{2}) = 1$? Commented Dec 28, 2016 at 0:40
• Also this is not really calculus or real-analysis, avoid putting non-applicable tags. Commented Dec 28, 2016 at 0:41
• @cool.coolcoolcool: I don't think I have any givens. Just the three identities. Commented Dec 28, 2016 at 0:43
• @de_dust That doesn't sound right. You must have been allowed to use the fact that $\sin(\pi/2) = 1$. Commented Dec 28, 2016 at 0:44
• It is impossible to answer the question without knowing for what $a$ the value $\sin a$ or $\cos a$ is assumed known.
– user9464
Commented Dec 28, 2016 at 0:44

In the sum of angle theorems, let $a=b$ so that

$$\cos(2a)=\cos^2(a)-\sin^2(a)$$

By the last identity, notice that

$$\cos^2(a)-\sin^2(a)=2\cos^2(a)-1$$

$$\cos^2(a)-\sin^2(a)=1-2\sin^2(a)$$

Now let $a=\pi/4$ and use known values.

• Perhaps one should add that "known values" here are assumed to be $\cos(\pi/2)=0$?
– user9464
Commented Dec 28, 2016 at 0:49
• @Jack One should figure that they need that as they are doing the problem, and it shouldn't be a super big deal (I'd really hope it isn't a problem) Commented Dec 28, 2016 at 0:50