Example 1:

Find the six trigonometric ratios of the angle in the following diagram:

Solution:
 sinO = __opposite sin = . ____________ sin0 = __hypotenuse sin.. =__2/3 csc0 = __hypotenuse csc = . ____________ csc0 = __opposite csc.. =__3/2 cos0 = __adjacent cos = . ____________ cos0 = __hypotenuse cos.. =__/3 sec0 = __hypotenuse sec = . ____________ sec0 = __adjacent sec.. =__3/ tanO = __opposite tan = . ____________ tan0 = __adjacent tan.. =__2/ cot0 = __adjacent cot = . ____________ cot0 = __opposite csc.. =__/2

Example 2:

If cos = 3/4, find the other five trigonometric ratios of .

Solution:
Since cos is defined as the ratio of the adjacent side to the hypotenuse, we drow a triangle with hypotenuse of length 4 and a side of length 3 adjacent to . We use the Pythagorean Theorem to find the opposite side. Let it be x for now. Then we get:
32 + x2 = 42
Therefore:
x2 = 7 so x =

Hence the triangle looks like:

________

So the other trig ratios are as follows:

 sin = /4 csc = 4/ sec = 4/3 tan = /3 cot = 3/