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
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.

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