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I was studying math.. and I just realized that I only just memorized these trigonometric equations, but I don't really know the reason behind them. So um...

Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$?

What happens if $x=3\theta$? Would the equations change to something like

$\cos(6\theta)=\cos^2(3\theta)-\sin^2(3\theta)$ and

$\sin(6\theta)=2\sin(3\theta)\cos(3\theta)$?

JimmyK4542
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user125342
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  • These are both immediate from the addition formulas for sine and cosine, which you can see proved here. In response to your second question, yes, both formulas are correct. –  Jul 05 '14 at 04:15
  • Yes, precisely. You may further reduce $\cos(3\theta)$ and $\sin(3 \theta)$ using other identities until you reach $\cos (\theta)$ and $\sin (\theta)$ or whatever else you may need. – Mark Fantini Jul 05 '14 at 04:16
  • With a little effort you can find, and test (maybe by induction) , a formula for $cos(n \theta)$ and for $sin(n\theta)$ using, e.g., Bonger's advice. – user99680 Jul 05 '14 at 04:21
  • Thank you everyone!!@bongers, i will study that now! – user125342 Jul 05 '14 at 04:29
  • Did you really mean to ask about the case $x = 3\theta$? That doesn't offer much insight, as you can see. I'm thinking you wanted to ask about $\cos 3x$ and $\sin 3x$. – M. Vinay Jul 05 '14 at 06:21

2 Answers2

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$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ Proof without words:

$\qquad\quad$

$\qquad\quad$ Image credit: "Blue's Blog: The Bloog!". See also this Math.SE answer by Blue.

Lucian
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  • Nice, based on pythagorean theorem directly (as indeed trigonometry is based on the pythagorean theorem). In my answer the pythagorean theorem is encoded inside the complex DeMoivre representation – Nikos M. Jul 05 '14 at 08:14
  • @Nikos: Where is Pythagoras here? – TonyK Jul 05 '14 at 09:25
  • Very nice - an image is worth more than 1000 words!!!! – johannesvalks Jul 05 '14 at 09:26
  • @TonyK, the pythagorean relation is first of all in the fact that $1 = (\cos \beta)^2 + (\sin \beta)^2$, the other right-angled triangles have same relations. Then by parallels and angles similarity the rest relations follow. Also $\cos \alpha = adjacent side / hypotenuse \implies adjacent side = \cos \alpha \times \cos \beta$ etc... (which is also related to pythagorean theorem) – Nikos M. Jul 05 '14 at 09:33
  • This should be the best answer... – johannesvalks Jul 05 '14 at 09:37
  • @Nikos: The fact that $\cos^2\beta + \sin^2\beta = 1$ is not used here. – TonyK Jul 05 '14 at 09:45
  • @Tony, equivalences between law of sines, law of cosines and pythagorean theorem are given is these notes (in case one would like to take a further look) – Nikos M. Jul 05 '14 at 09:59
  • @Tony, my comment is about the visual description of the identities using right-angled triangles (which relate directly to the pythagorean theorem, even though the relation $\cos^2\beta + \sin^2\beta = 1$ is not used explicitly but implicitly, yet the values of the sides stem and are related to it, as indeed ptolemy's theorem which is related can be proved by law of cosines and this in turn can be proved directly via pythagorean theorem). Your comment has a basis, i like to make these analogies myself, hope this is clearer now – Nikos M. Jul 05 '14 at 10:16
  • FYI: That's my image. See this answer and also my blog. – Blue Jul 05 '14 at 10:23
  • @Blue, personally already checked your blog post by following the link of the image – Nikos M. Jul 05 '14 at 10:26
  • @Lucian: You can't do that! I thought that you had made those lovely pictures. I would change my upvote to a downvote for plagiarism if I could. – TonyK Jul 05 '14 at 10:26
  • The fact that the image links to my site is appreciated; however, I don't usually expect to "click through" images here to see the source. An explicit statement of attribution would be more appropriate ... so I added one. (I could/should also watermark my blog images.) – Blue Jul 05 '14 at 10:40
  • @Blue: Actually, I found the image on Wikipedia $($a long time ago$)$, and then clicked on it and followed the link to its description page, where I also found the link to a site that had both together. If you're not cool with others using it, that's OK, just say the word, and I'll either find a suitable replacement by Google, or draw one myself using MicroSoft Visual Studio. Of course, it will take some time. – Lucian Jul 05 '14 at 14:30
  • @TonyK: I originally found it on Wikipedia a long time ago. Since it's public domain, I didn't think that there would be any problems. – Lucian Jul 05 '14 at 14:36
  • @Lucian: The Wikipedia contributor took (the idea of) the image there from the Math.SE post I linked above; he didn't originally give me credit, but a Math.SE-er later rectified that situation. (Hooray for the community!) Note that this stuff isn't "Public Domain". Math.SE (and the Wikipedia image) is under a Creative Commons license "with attribution required". (See the very bottom of any Math.SE page.) Please know that I think you acted in good faith by linking the image to my blog; there's no animosity here. An explicit credit is just better practice. I'm happy now. :) – Blue Jul 05 '14 at 14:54
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One way to see and also remember easier the always evolving trigonometric identities is using (a representation of) complex numbers and DeMoivre's theorem (or more correctly Euler's identity) like this:

$$e^{ix} = \cos(x) + i\sin(x)$$

Then

$$e^{i2x} = e^{ix} \times e^{ix}$$

$$\cos(2x) + i\sin(2x) = (\cos(x) + i\sin(x)) \times (\cos(x) + i\sin(x))$$

by equating real and imaginary parts respectively one gets the identities and so on..

Now this is purely formal as stated, but by actually taking into account the geometrical meaning (or representation) of a complex number (as is assumed in Euler's representation) this is made way more intuitive. For example multiplication of 2 complex numbers (of modulus $1$ i.e $e^{ia}$) has a direct meaning as geometrical rotation (on the unit circle) and the rest follow from that.

Furthermore, this representation of complex numbers of modulus one as representing rotations is further used in groups like $SU(n)$ and $U(1)$ which play a central part in physics and engineering (for example Quantum Mechanics)

Nikos M.
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