Chapter 5 Section 5.2 Trigonometric Functions

Angles in Standard Position Recall an angle in standard position is an angle that has its initial side on the positive x -axis. We can use any point on the angles terminal side to find the values of the trigonometric ratios. If the coordinates of the point P are ( x,y ) and the distance the point P is from the origin is r we get the following values for the trigonometric ratios. P :( x,y ) x -axis y -axis x y  In the example to the right with the coordinates of P at the point (1,3) P :( 1,3 )  r

It is important to realize that it does not matter what point you select on the terminal side of the angle the trigonometric ratios will be the same because the triangles are similar. The triangle with its vertex at P 1 is similar to the triangle with its vertex at P 2 and the length of the sides are proportional (equal ratios). P1P1 P2P2  Signs of Trigonometric Functions The trigonometric ratios now are defined no matter where the terminal side of the angle is. It can be in any if the four quadrants. Since the values for the xy -coordinates are different signs (±) depending on the quadrant the trigonometric ratios will be also. The value for r is always positive. The chart below shows the signs of the trigonometric ratios. x pos (+) y pos (+) x neg (-) y pos (+) x pos (+) y neg (-) x neg (-) y neg (-) Quadrantsincostancotseccsc I II III IV-+--+-

-3 2 r  Angle xy sincostancotseccsc undefined undefinedundefined 0 0undefined undefined1

Reciprocal Identities: Pythagorean Identities:

Second quadrant cosine is negative

In the third quadrant y is negative.