#### Trigonometric Functions

One of the easier ways to start understanding trigonometric functions is by picturing a right triangle. (Refer back to the triangles section to recall this.) Let be one of the acute angles. Then we will label the triangle as follows:

Now the trig. ratios can be defined for any acute angle
as follows:

opposite | adjacent | opposite | |||||

sin = | cos = | tan = | |||||

hypotenuse | hypotenuse | adjacent | |||||

--- | --- | ||||||

hypotenuse | hypotenuse | adjacent | |||||

csc = | sec = | cot = | |||||

opposite | adjacent | opposite |

- Note:
- The ratios are the same for any right triangle with angle , since when a triangle has equal angles they are similar trianges.

Now what if we have an angle greater than 90^{o} (ie. an obtuse angle)?
Well if we label the triangle like this:

(where r = SQRT(x

^{2}+ y

^{2}) by the Pythagorean Theorem)

Then sin = y/r, cos = x/r, tan = y/x, etc. Now we can extend the definition of trig. ratios to any angles. These have the same value for the trig. functions except for possibly a change of sign. (The comparable acute angle is known as the reference angle.) For instance in the triangle:

sin is the same as sin but cos = -cos .

From the note above, since the ratio is the same as long as the angles are the
same, let's assume that

When we substitute r = 1 into the equations, we get