Chapter 3 Trigonometric Functions of Angles Section 3.3 Trigonometric Functions of Angles

P:( x,y ) 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. 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-+--+-

Find the values of the six trigonometric functions if the point (-3,2) is on the terminal side of the angle. We find the value for r (distance from the origin) first r  Reference Angles The reference angle for an angle is the angle made when you drop a line straight down to the x -axis. it is the angle made by the x -axis regardless of what side of it you are on. 120  60  225  45  330  30  -300  60 

Reference angles are useful to help you find the values for trigonometric functions for many angles of the circle. For example if we want to find the trigonometric ratios for 150 . We know the reference angle is 30  and we for a 30  -60  -90  triangle. The sides are in the ratios we mentioned before. 150  30  1 Identities