Trigonometric Functions
2.2 – Definition 2
JMerrill, 2006
Revised, 2009
(contributions from DDillon)
Angle of Elevation Review
• The angle of elevation of from the ground to the
top of a mountain is 68o. If a skier at the top of
the mountain is at an elevation of 4,200 feet, how
long is the ski run from the top to the base of the
mountain?
• 4,529.85 feet
Navigation Review
• If a plane takes off on a heading of N 33o W and
flies 12 miles, then makes a right (90o) turn, and
flies 9 more miles, what bearing will the air
traffic controller use to locate the plane? How
far is the plane from where it started?
• The plan is 15 miles away on a bearing of N 8.41o E
Defining Trig Functions
Let there be a point P (x, y) on a
coordinate plane.
P(x, y)
y
r
θ
x
r is the distance from the origin
to the point P, which can be
represented as being on the
terminal side of θ. Since r
represents a distance, it is
always positive and cannot = 0.
The six trigonometric functions
are:
Definition 2
y
sin  
r
x
cos  
r
y
tan  
x
r
csc 
y
r
sec 
x
x
cot  
y
Remember, the denominator cannot ever = 0!
Calculating Trig Values for
Acute Angles
If the terminal side of θ in standard position passes through
point P (6, 8), draw θ and find the exact value of the six trig
functions of θ.
r 10 5
y 8 4
csc    
sin    
y 8 4
r 10 5
P(6, 8)
r = 10
8
x 6 3
cos   

r 10 5
sec  
r 10 5
 
x 6 3
y 8 4
tan    
x 6 3
cot  
x 6 3
 
y 8 4
r is the hypotenuse and can be
found using Pythagorean Thm:
θ
6
x2 + y2 = r2
You Do
• If the terminal side of θ in standard position
passes through point P (3, 7), draw θ and find the
exact value of the six trig functions of θ.
y 5 58
sin   
r
58
csc  
r  58

y
5
x 3 58

r
58
sec  
r  58

x
3
cos  
tan  
y 5

x 3
cot  
x 3

y 5
Calculating Trig Values for
Nonacute Angles
If the terminal side of θ in standard position passes through
point P (-4, 2), draw θ and find the exact value of the six trig
functions of θ.
r 2 5
y
2
5
csc   
 5
sin   

y
2
r 2 5
5
cos  
P(-4, 2)
2
r = 2√5
θ
-4
x
4
2 5


r 2 5
5
y 2
1
tan   

x 4
2
sec  
r 2 5
5


x
4
2
cot  
x 4

 2
y 2
Calculating Trig Values for
Quadrant Angles
•
Find the exact value of the six trig functions when θ=90o
•
A convenient point on the terminal side of 90o is (0,1). So x = 0,
y = 1, r = 1
y 1
= =1
r 1
x 0
cos   = =0
r 1
y 1
tan   = =U
x 0
sin  
(0,1)
Now, if the angle is 180o, what
point will you use?
r 1
 1
y 1
r 1
sec     U
x 0
x 0
cot     0
y 1
csc  