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When simplifying a trigonometric expression, say, $\sin^2 \theta$ / $\cos^2 \theta$ - I remember that sin over cos is equal to tan.

However, what other identities, such as the one mentioned above, do I need to know in general?

Is their a way to quickly work them out? or do you have to memorize them?

Thanks and regards,

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    $\begingroup$ What you're supposed to do is do so many problems that along the way you learn everything you'll ever need. $\endgroup$ – Gerry Myerson Jul 21 '14 at 13:16
  • $\begingroup$ @GerryMyerson I agree, so basically I need to memorize that identity, and the ones related to it aswell? $\endgroup$ – sasha Jul 21 '14 at 13:19
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    $\begingroup$ That's not an identity, that's a definition. :) (I suppose technically, it is also an identity, but...) $\endgroup$ – Thomas Andrews Jul 21 '14 at 13:19
  • $\begingroup$ There are certain ways to quickly re-work the formulas for angle summation - $\sin(A+B), \cos(A+B)$, etc. $\endgroup$ – Thomas Andrews Jul 21 '14 at 13:21
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    $\begingroup$ Let me elaborate. I don't recommend sitting down with a list of things to memorize. I recommend doing a lot (a LOT) of problems. When you get stuck in a problem, look for some identity or definition that will help you solve that problem. The more problems you do, the less you'll find yourself needing to look things up --- you will have "memorized" the important stuff without specifically trying to memorize anything, and you will have internalized which formulas are most likely to work in which situations. $\endgroup$ – Gerry Myerson Jul 21 '14 at 23:51
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Here is an exercise that may help you out. enter image description here

The top triangle is actually 4 similar triangles being overlaid. On the bottom is a pink triangle; on top of that is an orange triangle, a beige one, and a green one. Below the big triangle I have shown the smaller triangles in isolation.

Your task is 3-fold.

  1. find the length of the unknown sides in the orange, beige, and green triangles (express them in terms of trigonometric functions of $t$)
  2. write out the definitions of each trigonometric function (cos, sin, tan, csc, sec, cot) using each of the 4 triangles (that's 4 sets of definitions).
  3. write out the Pythagorean Theorem for each of the 4 triangles.
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  • $\begingroup$ And once you remember those three triangles, very often you can draw one of them with its sides appropriately labeled to find an instant answer to a problem (or to one of the steps in solving a problem). Sometimes you have to multiply all three sides by the same amount after drawing the triangle; that doesn't change the angle. I learned this tool more than 30 years ago and still use it sometimes. $\endgroup$ – David K Jul 22 '14 at 1:51
  • $\begingroup$ @David K Thank you for your comment. There is a 4th Pythagorean identity that is sometimes useful in certain situations (e.g. simplify $\tan\theta\csc\theta$). If we take $\sin^2\theta+\cos^2\theta=1$ and multiply both sides by $\sec^2\theta\csc^2\theta$ we end up with the identity $\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$. I use this triangle to remember that $\tan\theta = \sec\theta/\csc\theta$. $\endgroup$ – John Joy Jul 22 '14 at 12:44
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There are several useful identities. However these basic once you remmember as soon as you understand what it actually means.

$ \cos t =\frac{\text{adj}}{\text{hyp}}$

$ \sin t =\frac{\text{opp}}{\text{hyp}}$

$ \tan t =\frac{\sin }{\cos }$=$\frac{\text{opp}}{\text{adj}}$

enter image description here

We know from the phytagorean Theorem Identity that $\sin ^2(x)+\cos ^2(x)=1$ By dividing by $\sin ^2(x)$ or $\cos ^2(x)$ we are getting two useful alternative versions of that identity.

Other useful identities are double angle identities, you are not expected to remember all identities. However some of them you end up using so often that they become second nature.

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