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 theta 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 theta as follows:

opposite adjacent opposite
sin theta =
cos theta =
tan theta =
hypotenuse hypotenuse adjacent
--- ---
hypotenuse hypotenuse adjacent
csc theta =
sec theta =
cot theta =
opposite adjacent opposite
Note:
The ratios are the same for any right triangle with angle theta, 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 90o (ie. an obtuse angle)? Well if we label the triangle like this:


(where r = SQRT(x2 + y2) by the Pythagorean Theorem)

Then sin theta = y/r, cos theta = x/r, tan theta = 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 theta is the same as sin alpha but cos theta = -cos alpha.

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


Introductory Exercises Moderate Exercises
menu