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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.

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

1 Answer 1

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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.

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  • $\begingroup$ Perhaps one should add that "known values" here are assumed to be $\cos(\pi/2)=0$? $\endgroup$
    – user9464
    Commented Dec 28, 2016 at 0:49
  • $\begingroup$ @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) $\endgroup$ Commented Dec 28, 2016 at 0:50

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