Download Sec 6.5

6.5 Trigonometric Functions of General Angles
OBJECTIVE 1: Understanding the Four Families of Special Angles
The Quadrantal Family of Angles
The
The
The

3

6

4
Family of Angles
Family of Angles
Family of Angles
OBJECTIVE 2: Understanding the Definitions of the Trigonometric Functions of General Angles
opp y

hyp r
adj x
cos  

hyp r
opp y
tan  

adj x
sin  
hyp r

opp y
hyp r
sec  

adj x
adj x
cot  

opp y
csc  
The General Angle Definition of the Trigonometric Functions
If P ( x, y ) is a point on the terminal side of any angle  in standard position and if r  x 2  y 2 is
the distance from the origin to point P, then the six trigonometric functions of  are defined as follows:
y
r
x
cos  
r
y
tan   , x  0
x
sin  
r
, y0
y
r
sec   , x  0
x
x
cot   , y  0
y
csc  
OBJECTIVE 3: Finding the Values of the Trigonometric Functions of Quadrantal Angles
OBJECTIVE 4: Understanding the Signs of the Trigonometric Functions
II
I
Sine (and cosecant) are positive
All trigonometric functions are positive
Tangent (and cotangent) are positive
Cosine (and secant) are positive
III
IV
OBJECTIVE 5: Determining Reference Angles
Definition
Reference Angle
The reference angle,  R , is the positive acute angle associated with a given angle  . The
reference angle is formed by the “nearest” x-axis and the terminal side of  . Below are angles
sketched in each of the four quadrants along with their corresponding reference angles.
Quadrant I
Quadrant III
Quadrant II
Quadrant IV
The measure of the reference angle  R depends on the quadrant in which the terminal side of C lies.
The four cases are outlined below.
Quadrant II
Quadrant I
Case 1: If the terminal side of
lies in Quadrant I, then
C
 R  C .
Case 2: If the terminal side of
lies in Quadrant II then
C
 R    C
Quadrant III
R  180  C .)
Quadrant IV
Case 3: If the terminal side of
lies in Quadrant III then
(or
C
 R  C  
Case 4: If the terminal side of
(or
R  C 180
.)
in Quadrant IV then
C lies
 R  2  C
(or
R  360  C .)
OBJECTIVE 6: Evaluating Trigonometric Functions of Angles Belonging to the
Steps For Evaluating Trigonometric Functions of Angles Belonging to the
 

,
, or
Families
3 6
4
 

, , or
Families
3 6
4
Step 1: Draw the angle and determine the quadrant in which the terminal side of the angle lies.
Step 2: Determine if the sign of the function is positive or negative in that quadrant.
 

, , or .
3 6
4
Step 4: Use the appropriate special right triangle or your knowledge of Table 1 to determine
the value of the trigonometric function.
Step 3: Determine if the reference angle,  R , is