13.3A Trigonometric Functions of
Any Angle
Algebra II
Trig functions
Sin θ =
Cos θ =
Tan θ =
y
r
x
r
y
x
Csc θ =
Sec θ =
Cot θ =
r
y
r
(x,y)
x
X
y
r = √x² + y²
r
θ
Ex.) 1
Use the pythagorean theorem to find the value of r. Let (-5,12) be a point on the
terminal side of an angle θ in standard position. Then evaluate the six
trigonometric functions of θ.
r = √(-5)² + (12)²
Sin θ = Y =
r
Csc θ = R =
y
Cos θ = X =
r
Sec θ = R =
x
y
Tan θ = x =
Cot θ =
X
y
=
Quadrantal Angle
• When the terminal side of θ lies on an axis, it is
called a quadrantal angle.
(0,r)
Θ=0
θ
(r,0)
θ
θ
(0,-r)
(-r,0)
Unit Circle
• The circle x 2  y 2  1 which has center (0,0) &
radius of 1, is called the unit circle. The values
of sin  & cos  are simply the y-coordinate
& x-coordinate, respectively of the point
where the terminal side of  intersects the
unit circle.
Ex.) 2
Evaluate the six trigonomic functions
When θ =
 , then x =____ and y = ____.
sin θ =
=
csc θ =
=
cos θ =
=
sec θ =
=
tan θ =
=
cot θ =
=
13-3B Finding Reference Angles
• Reference Angle for  is the acute angle  
formed by the terminal side of  & the x-axis.
Steps to Find Reference Angles
• 1)Find the quadrant angles that it is between.
90    180

 
2
Degrees :  '  180  
Radians :  '    
180    270
270    360
3
2
Degrees :  '    180
3
   2
2
Degrees :  '  360  
Radians :  '    
Radians :  '  2  
  
• 2)Then, follow the correct subtraction process
above.
Ex.) 3
Find the reference angle θ’ for angle θ.
a.) θ = 140°
c.) θ = -295°
3
b.)  
4
d.)

7
12
Using Reference Angles to Evaluate
Trigonometric Functions
• First find the reference angle
• Do the given operation to the reference angle
from Signs of Function Values
• Quadrant II
Quadrant I
sin  , csc  : 
cos , sec : tan  , cot : -
• Quadrant III
sin  , csc  : 
cos , sec : tan  , cot : 
sin  , csc : 
cos , sec : 
tan  , cot : 
Quadrant IV
sin  , csc : cos , sec : 
tan  , cot : -
Ex.4)
Evaluate a.) sin (-225°)
5

b.) cot
3