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
|sin =||cos =||tan =|
|csc =||sec =||cot =|
- 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 90o (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
When we substitute r = 1 into the equations, we get