In mathematics the sine, cosine and tangent of an acute angle in a right-angled triangle are defined in terms of the sides of the triangle as follows:

________________________________________________________________________
|                                                                       |
| dgm 1    |\                      sin = opposite/hypotenuse  : S = O/H |
|          | \                                                          |
|          |  \                    cos = adjacent/hypotenuse  : C = A/H |
|          |   \                                                        |
|          |    \                  tan = opposite/adjacent    : T = O/A |
|  Opposite|     \ Hypotenuse                                           |
|          |      \                                                     |
|          |       \                                                    |
|          |        \                                                   |
|          |        /\                                                  |
|          |_______|_x\                                                 |
|            Adjacent                                                   |
|                                                                       |
|_______________________________________________________________________|

An easy method to remember dgm 1 is: SOH.CAH.TOA

The sine, cosine and tangent of 30⁰, 45⁰ and 60⁰, can be expressed exactly in surd form and are worth remembering.

________________________________
|                               |
| dgm 2    |\                   |
|          | \                  |
|          |  \                 |
|          |   \                |
|          |    \    __         |
|        1 |     \  \|2         |
|          |      \             |
|          |       \            |
|          |        \           |
|          |        /\          |
|          |_______|45\         |
|                               |
|                1              |
|_______________________________|

The triangle in dgm 2 shows the following:

sin 45⁰ = 1/√2

cos 45⁰ = 1/√2

tan 45⁰ = 1

________________________________
|                               |
| dgm 3       /|\               |
|            / | \              |
|           /  |30\             |
|          /   |---\            |
|         /    |    \           |
|        /     |     \  2       |
|       /  __  |      \         |
|      /  \|3  |       \        |
|     /        |        \       |
|    /\        |        /\      |
|   /60|_______|_______|60\     |
|                    1          |
|_______________________________|

The triangle in dgm 3 shows the following:

sin 60⁰ = √3/2

cos 60⁰ = 1/2

tan 60⁰ = √3

sin 30⁰ = 1/2

cos 30⁰ = √3/2

tan 30⁰ = 1/√3

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Note - The standard way to measure angles in mathematics is anticlockwise from the positive-X direction. But I did it the other way for ease of ascii art...
Thank you hobyrne for pointing that out

Silly Old Harry Caught A Herring Trawling Off America.

It is also worth knowing that

        sin x
tan x = -----
        cos x

You can visit trigonometric identities for many, many more similar mind-numbingly exciting pieces information. Don't laugh, some of this stuff is actually pretty useful.

Three additional, somewhat redundant trigonometric ratios are found by taking the reciprocals of the first three.

Secant

          1     hypotenuse
sec x = ----- = ----------
        cos x    adjacent

Cosecant

          1     hypotenuse
csc x = ----- = ----------
        sin x    opposite

Cotangent

          1      adjacent
cot x = ----- = ----------
        tan x    opposite

It's not recommended to use these for the purposes of trying to rearrange trig formulae. Having both sine, cosine and tangent and their reciprocals in the same equation can be terribly, terribly confusing and lead to errors. Stick wholly with sines and cosines (and tangents if you like, but they can knocked out, see above) until right at the end when you want to make the final expression as small as possible.

The terms secant, cosecant and cotangent for the reciprocals of sine, cosine and tangent are, as far as I can tell, only kept in use i) for completeness, and ii) because they are shorter to say and write under certain circumstances. They allow us to say for instance that

d
-- tan x dx = sec x tan x
dx

which is an elegant way of expressing a standard definition, as opposed to the relatively long-winded

d               sin x
-- tan x dx = --------
dx             cos2 x

which is something of a mouthful to pronounce. Finally, never confuse the following

(sin x)-1 and sin-1 x

(cos x)-1 and cos-1 x

(tan x)-1 and tan-1 x

On the left, trigonometric functions inverted; on the right, inverse trigonometric functions. NEVER confuse the two. No good will come of this. See also hyperbolic functions.

Trigonometric Ratios

There are three basic ratios in trigonometry (All assume a Right Triangle):

• Sine: the sine is the ratio of the side opposite to a given angle to the hypotenuse
• Cosine: the cosine is the ratio of the side adjacent to a given angle to the hypotenuse
• Tangent: the tangent is the ratio of the side opposite to a given angle to the adjacent side of the angle.

Sine is abbreviated Sin; Cosine is cos; and Tangent is tan.

OK. So what does that mean? That means that if you have a right triangle (ΔABC--I can't find an HTML triangle symbol, so Delta will have to do will have to do) and an angle (∠A), you have three sides:

    A
      |\
      | \       AB=20
  a   |  \      BC=19 
  d   |   \     AC=√(20² + 19²)≈27.5862
  j   |    \             
  a   |     \ 
  c   |      \
  e   |       \ 
  n   |        \     
  t   |         \                 opposite to T
      |          \                      |           RS=7
  t   |           \ hypotenuse          |           ST=6
  o   |            \                    | R         AC=√(7² + 6²)≈9.2195
      |             \                   |  |\
  A   |              \                  |  | \
      |               \                 |  |  \
      |                \                 ->|   \ hypotenuse
      |                 \                  |    \
      |_                 \                 |_    \
      |_|_________________\                |_|____\
     B    opposite to A    C              S   ^    T
                                              |________ adjacent to T    

The opposite is the leg that doesn't touch the angle. BC doesn't touch ∠A, so it is opposite to ∠A. The hypotenuse is the side opposite to the right angle of the triangle. In our triangle ΔABC, the hypotenuse is AC. The adjacent side is the leg that touches the angle (but isn't the hypotenuse!). In ΔABC, AB is the adjacent side, because it touches ∠A, but isn't the hypotenuse.

Now, let's get back to the ratios. In ΔABC, the sine of ∠A would be the length of BC over AC. Thus: Sin A = (19/27.5862) ≈ 0.6887. ∠A's cosine would be AB over AC, or 20/27.5862 ≈ 0.7250. The tangent of ∠A would be BC over AB, or 19/20 = .95.

Easy way to remember: Some old hippy caught another hippy tripping on acid.

Arcsine, Arcosine, and Arctangent

Arcsine, arcosine, and arctangent are the inverse of sine, cosine, and tangent. Where sin, cos, and tan give ratios when given an angle, arcsine, arcosine, and arctangent (abbreviated sin^-1, cos^-1, and tan^-1 respectively) take a ratio and give you an angle measurement.

If we wanted to find the measure of ∠A and we have the tangent, we would take tan^-1(95) ≈ 44°. If we wanted to find m∠C, we would take sin^-1(sin(C)) = sin^-1(AB/AC) = sin^-1(.7520) ≈ 46°

The nice part of this is that since all the interior angles in a triangle add up to 180°, 180=90+44+46 and thus, 180=180, meaning my ASCII triangles up there actually came out right. (BAD JOKE AHEAD!) I sure went off on a tangent there. (ACK! THERE IT IS!)

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Hooray for highschool math! Tell me if I screwed something up.