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.

Let's take a look at some examples

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(x² + y²) 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 r = 1. We can now place the triangle on the unit circle with O at the center of the circle and P as a point on the circle. Here's a diagram to see what we mean.

When we substitute r = 1 into the equations, we get sin = y and cos = x. Therefore, that means that the x-coordinate of the point P gives the value of cos and the y-coordinate of the point P gives the value of sin . From sin and cos , we can figure out the remaining trig. ratios. Here are some examples using trig. functions.

---