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This is a very straight-forward question:

I'm trying to simplify $$\sin(2\pi t +\pi/4) + \sin(2\pi t -\pi/4)$$ and failing at it:

$\sin(2\pi t +\pi/4) + \sin(2\pi t -\pi/4)$
$2\sin(2\pi t)\cos(\pi/2)$ by sum $\to$ product = ZERO

Wolfram alpha gives $\sqrt{2}\sin(2\pi t)$, but gives no explanation as to how.

What am I doing wrong?

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What does the -20t mean? – Ted Feb 27 '13 at 6:54
@Ted forgot to omit that irrelevant part of the question. Fixed – Griffin Feb 27 '13 at 20:15
I don't know why this got so many votes... It was just a small error in arithmetic. – Griffin Feb 27 '13 at 20:25
You forgot to divide $\pi/2$ by 2. Doing this gives $\cos(\pi/4) = 1/\sqrt{2}$ instead of $0$, which gives the right answer. – Joe Z. Feb 27 '13 at 21:11
up vote 10 down vote accepted

If you draw the situation on the unit circle, then $\sin\varphi$ is the x-coordinate of the point corresponding to the angle $\varphi$. The points $2\pi t-\pi/4$ and $2\pi t+\pi/4$ have right angle between them and $2\pi t$ is exactly in the middle.

So from the right triangle below you see that the sum of the vectors corresponding to $2\pi t-\pi/4$ and $2\pi t+\pi 4$ has the same direction as the vector corresponding to $2\pi t$ and the length is $\sqrt2$. The sum $\sin(2\pi t-\pi/4)+\sin(2\pi t+\pi/4)$ is the x-coordinate of this vector, and it is equal to $\sqrt2\sin 2\pi t.$

The same argument gives $\cos(2\pi t-\pi/4)+\cos(2\pi t+\pi/4)=\sqrt2\cos 2\pi t.$

unit circle

More-or-less the same idea can be rewritten using complex numbers if we use Euler's forumula $e^{i\varphi}=\cos\varphi+i\sin\varphi$.

We have $$e^{\alpha+\pi/4}+e^{\alpha-\pi/4}=e^\alpha(e^{\pi/4}+e^{-\pi/4})=e^\alpha2\cos\frac\pi4=e^\alpha\sqrt2.$$ The real part gives $\cos(\alpha+\pi/4)+\cos(\alpha-\pi/4)=\sqrt2\cos\alpha$ and the imaginary part gives $\sin(\alpha+\pi/4)+\sin(\alpha-\pi/4)=\sqrt2\sin\alpha$.

Complex numbers are quite often useful for remembering/proving trigonometric identities.

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v. v. thorough! – bharal Feb 27 '13 at 14:00
I like this because it's so visual. Accepted! – Griffin Feb 27 '13 at 20:29

Hint: $$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$$

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why do this instead of using the sum->product formula? – Griffin Feb 27 '13 at 20:26

If you want to use the sum-product formula, it should be the right one. We have $$\sin x+\sin y=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right).$$ But I think it is better to go back to the addition/subtraction laws for sine, as described by Zev Chonoles.

We have $$\sin(a+b)=\sin a\cos b+\cos a\sin b,$$ and its very close relative $$\sin(a-b)=\sin a \cos b -\cos a\sin b.$$ Let $a=2\pi t$ and $b=\frac{\pi}{4}$. The sine and cosine of $\frac{\pi}{4}$ are both $\frac{1}{\sqrt{2}}$.

Remark: If one will be doing a certain type of calculation very often, it is useful to commit to memory any relevant formulas. However, it is I think best to know only a quite small number of trigonometric facts, and reconstruct anything else one may need.

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We keep going back and forth :) I just deleted my answer to simplify everything, and besides yours is great as always. – Zev Chonoles Feb 27 '13 at 7:12
@Zev Chonoles: I don't see the point of deleting. If you undelete, I can upvote. There is much to be said for a minimalist nudge in the right direction. – André Nicolas Feb 27 '13 at 7:15

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