Download 13.3 Trigonometric Functions of Any Angle

13.3 Trigonometric Functions
of Any Angle
Algebra 2
General Definition of Trigonometric
Functions

Let θ be an angle in standard position and (x,
y) be any point (except the origin) on the
terminal side of θ. The six trigonometric
functions of θ are defined as follows.
y
sin  
r
x
cos  
r
y
tan   , x  0
x
r
csc   , y  0
y
r
sec   , x  0
x
x
cot   , y  0
y
Examples:
1)
Let (-4, -3) be a point on the terminal side of
θ. Evaluate the six trigonometric functions of
θ.
2)
Let (-5, 12) be a point on the terminal side
of θ. Evaluate the six trigonometric
functions.
Example:
Use the given point on the terminal side of
an angle θ in standard position. Evaluate
the six trigonometric functions of θ.

1)
2)
(4, 5)
3
7,  2

Quadrantal Angles

Quadrantal angles- angles whose terminal
side of θ lies on an axis.
0 or 0 radians
90 or

2
radians
Quadrantal Angles (Continue)
180 or  radians
3
270 or
radians
2
Examples:
1)
2)
Evaluate the six trigonometric functions of θ
= 270˚.
Evaluate the six trigonometric functions of θ
= 90˚.
Reference Angles

When wanting to determine the trigonometric
ratios for angles greater than 90˚ (or less
than 0˚) must use corresponding acute
angles.

Reference Angles: (corresponding acute
angles) an acute angle θʹ formed by the
terminal side of θ and the x-axis.
Reference Angle Relationships
90    180

2
 
Degree :  '  180 - 
Radians :  '    
180    270
  
3
2
Degrees :  '    180
Radians :  '   - 
Reference Angles Relationships
270    360
3
   2
2
Degree :  '  360  
Radians :  '  2  
Examples:

Find the reference angle θʹ for each angle θ.

  140

  250

3
 
4

8

3
Evaluating Trigonometric Functions.
Step for evaluating a
trigonometric function
of any θʹ.

1)
2)
3)
Find the reference
angle, θʹ.
Evaluate the
trigonometric function for
the angle θʹ.
Use the quadrant in
which θ lies to determine
the sign of the
trigonometric function.
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
Examples:

Evaluate…
 cos150



sin 225
sin  390
sec120
Examples:

Evaluate…

 7 
cos

 4 

 7 
sec

 6 

 16 
cot 

 3 

 2 
sin  

 3 
Examples:

v2
Using the formula, d  sin 2 estimate the
32
horizontal distance traveled by a golf ball hit
at an angle of 40˚ with an initial speed of 125
feet per second.
Example

A golf club called a wedge is made to lift a
ball high in the air. If a wedge has a 65˚ loft,
how far does a ball hit with an initial speed of
100 feet per second travel?
Example:

Your marching band’s flag corps makes a
circular formation. The circle is 20 feet wide
in the center of the football field. Our starting
position is 140 feet from the nearer goal line.
How fare from this goal line will you be after
you have marched 120˚ counterclockwise
around the circle?
Example:

A circular clock gear is 2 inches wide. If the
tooth at the farthest right edge of the gear
starts 10 inches above the base of the clock,
how far above the base is the tooth after the
gear rotates 240˚ counterclockwise?