Trigonometric Functions of any Angle

When evaluating any angle θ , in standard position, whose terminal side is given by the coordinates (x,y), a reference angle is always used. Notice how a right triangle has been created. This will allow us to evaluate the six trigonometric functions of any angle.
Notice the side opposite the angle θ has a length of the y value of the given coordinates. The adjacent side has a length of the x value of the coordinates. The length of the hypotenuse is given by

x2 + y2 .

Lets say, for the sake of argument, the length of the hypotenuse is 1 unit. This would mean the following would be true.

1 y 1 cos θ = x sec θ = x y x tan θ = cot θ = x y
You must think of the sine function as giving you the y value, whereas the cosine function yields the x value. This is how we will determine whether the sine, cosine, tangent, cosecant, secant or cotangent of a given angle is a positive or negative value. sin θ = y

csc θ =

If the angle to be evaluated is in quadrant IV, for instance, the sine of the angle θ will be negative. The cosine of θ , in this instance, will be positive, while the tangent of the angle θ will be negative.

Example
Evaluate the six trigonometric functions of an angle θ , in standard position, whose terminal side has an endpoint of (-3,2).

The angle with terminal side is first drawn.
Remember, in order to evaluate the six trigonometric
'

functions for θ , use the reference angle θ .

From the endpoint of the terminal side of the angle, a line is drawn to the x axis. This is the reason reference angles are always drawn in relation to the x axis. It will always create a right triangle with which to work.
Now all that is needed to solve the problem, is to find the length of the hypotenuse then the values of the six trigonometric functions can be found.

Using the