Lesson 5-3

Example 1
Use the unit circle to find each value.

a.	tan 180°

	The terminal side of a 180° angle in standard	
position is the x-axis, which intersects the
unit circle at (-1, 0). The x-coordinate of this
ordered pair is cos 180°, and the y-coordinate
is sin 180°. Since tan  = , tan 180° = 
or , which is 0. Therefore, tan 180° = 0.

b.	csc (-90°)

	The terminal side of a -90° angle in standard	
position is the negative y-axis, which
intersects the unit circle at (0, -1).
According to the definition of cosecant,
csc (-90°) =  or , which is -1. Therefore,
csc (-90°) = -1.

Example 2
PHYSICS  In physics, work is a force,	
measured in joules, that causes the
displacement of an object. The
equation for work W is W = F  s cos ,
where F is the force applied to an
object, s is the displacement, and  is
the angle between the force and the
displacement. A force of 150 Newtons
is applied to a box to slide it 10 meters
across a floor at a constant speed. If the
angle  between the applied force and
the displacement is between 0° and 90°,
what is the maximum amount of work
done by the applied force?

If the force applied and the horizontal	
displacement are both going toward the
right parallel to one another, the angle 
between the force and displacement would
be 0°. Find the work done for  = 0°.

W = F  s cos 
W = 150(10) cos 0°
W = 1500 J

If the force applied to the box is directly	
upward, the angle  between the applied
force and the displacement is 90°. Find
the work done for  = 90°.

W = F  s cos 
W = 150(10) cos 90°
W = 0 J

The maximum amount of work done by
the applied force is between 0 Joules and
1500 Joules.

Example 3
Use the unit circle to find the values of the six trigonometric functions for a 300° angle.

Since 300° is between 270° and 360°, the	
terminal side is in the fourth quadrant.
Therefore, the reference angle is
360° - 300° or 60°. The terminal side of
a 60° angle intersects the unit circle at
coordinates , . Because the
terminal side of a 300° angle is in the
fourth quadrant, the x-coordinate is
positive, and the y-coordinate is negative.
The point of intersection has coordinates , -.

sin (300°) = y
sin (300°) = -	cos (300°) = x
cos (300°) = 	tan (300°) = 
tan (300°) = 
tan (300°) = -	
			
csc (300°) = 
csc (300°) = 
csc (300°) = -
csc (300°) = -	sec (300°) = 
sec (300°) = 
sec (300°) = 2	cot (300°) = 
cot (300°) = 
cot (300°) = - 
cot (300°) = -	

Example 4
Find the values of the six trigonometric	
functions for angle  in standard position
if a point with coordinates (-6, 8) lies on
its terminal side.

You know that x = -6 and y = 8. You need to
find r.
r = 	Pythagorean Theorem
r = 	Substitute -6 for x and 8 for y.
r =  or 10	Disregard the negative root.

Now write the ratios.

sin  = 
sin  =  or 	cos  = 
cos  =  or -	tan  = 
tan  =  or - 	
			
csc  = 
csc  =  or 	sec  = 
sec  =  or -	cot  = 
cot  =  or - 	

Example 5
Suppose  is an angle in standard position whose terminal side lies in Quadrant II.
If csc  = , find the values of the remaining five trigonometric functions of .

To find the other function values, you must	
find the coordinates of a point on the terminal
side of . Since csc  = , r = 8 and y = 3.

Find x.

	r2	= x2 + y2	Pythagorean Theorem
	82	= x2 + 32	Substitute 8 for r and 3 for y.
	64	= x2 + 9
	55	= x2
		= x	Take the square root of each side.

Since the terminal side of   lies in Quadrant II, x must be negative.
Thus, x = -.

Now use the values of x, y, and r to find the remaining five trigonometric functions of .

sin  = 
sin  = 		
		
cos  = 
cos  =  or -	tan  = 
tan  =  or -	
		
sec  = 
sec  =  or -	cot  = 
cot  =  or -