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.

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!)

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