Definition of Trigonometric Functions

• With trigonometric ratios of acute angles in triangles, we are limited to angles between 0 and 90 degrees.

• We now define the trigonometric functions in terms of angles in standard form in the coordinate plane.

• Definition: Let an angle x (in radian measure) be in standard position. Let a point P with coordinates (a,b) be the point of intersection of the terminal side of the angle and the circle of radius r (were r > 0). Then:

sinb

xr

cosa

xr

tan , 0b

x aa

• Definition: Let an angle x (in radian measure) be in standard position. Let a point P with coordinates (a,b) be the point of intersection of the terminal side of the angle and the circle of radius r (were r > 0). Then:

csc , 0r

x bb

sec , 0r

x aa

cot , 0a

x bb

• Note that if the radius of the circle is 1 we have the following:

sinb

xr

cosa

xr

1

b b

1

a a

• We call the circle of radius 1 the unit circle, and on the unit circle we have …

( , )a b (cos ,sin )x x

Example 1:Find the values of the six trigonometric functions for angle x with the terminal side containing the point ( −3,−5).

3, 5a b

2 23 5

• The distance from the given point to the origin (0,0), which is radius r, is given by …

9 25 34

2 2r a b

3, 5, 34a b r

sinb

xr

cosa

xr

tanb

xa

5

34

5

3

3

34

5

3

3, 5, 34a b r

34

5

3

5

3

5

cscr

xb

secr

xa

cota

xb

34

5

34

3

34

3

Example 2:Let x have a terminal side containing point (a,b), where the point is in quadrant II, and sin x = 2/7.

sinb

xr

2

7 2, 7b r

2 2 2a b r 2 2 22 7a 2 4 49a 2 45a

3 5a

• Since the point is in quadrant II, the value of a must be negative, and we have …

3 5a

3 5a

3 5, 2, 7a b r

sinb

xr

cosa

xr

tanb

xa

2

7

2

3 5

3 5

7

2

3 5

7

2

3 5

2

cscr

xb

secr

xa

cota

xb

7

3 5

3 5, 2, 7a b r

7

3 5