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Trigonometric Functions:
The Unit Circle
&
Right Triangle Trigonometry
Quick Review
Give the value of the angle  in degrees.
2
1.  
3
2.   

4
Use special triangles to evaluate.
 
3. cot   
 4
 7 
4. cos  

6


5. Use a right triangle to find the other five trigonometric
4
functions of the acute angle  given cos  
5
Quick Review Solutions
Give the value of the angle  in degrees.
2
1.  
120
3
2.   

 45
4
Use special triangles to evaluate.
 
3. cot     1
 4
 7 
4. cos  
  3/2
 6 
5. Use a right triangle to find the other five trigonometric
4
functions of the acute angle  given cos  
5
sec   5 / 4, sin   3 / 5, csc   5 / 3, tan   3 / 4, cot   4 / 3
What you’ll learn about
• How to identify a Unit Circle and its
relationship to real numbers
• How to evaluate Trigonometric Functions
using the unit circle
• Periodic Functions of sine and cosine
functions
… and why
Extending trigonometric functions beyond
triangle ratios opens up a new world of
applications to model and solve real-life
problems.
Initial Side, Terminal
Side
Positive Angle, Negative
Angle
of a Nonquadrantal Angle
θ
1.
2.
3.
4.
5.
Draw the angle θ in standard position, being
careful to place the terminal side in the correct
quadrant.
Without declaring a scale on either axis, label a
point P (other than the origin) on the terminal side
of θ.
Draw a perpendicular segment from P to the xaxis, determining the reference triangle. If this
triangle is one of the triangles whose ratios you
know, label the sides accordingly. If it is not, then
you will need to use your calculator.
Use the sides of the triangle to determine the
coordinates of point P, making them positive or
negative according to the signs of x and y in that
particular quadrant.
Use the coordinates of point P and the definitions
to determine the six trig functions.
Trigonometric Functions of
any Angle
Let  be any angle in standard position and let P( x, y ) be any point on the
terminal side of the angle (except the origin). Let r denote the distance from
P ( x, y ) to the origin, i.e., let r  x  y . Then
2
y
r
x
cos  
r
y
tan  
( x  0)
x
sin  
r
y
r
sec  
x
x
cot  
y
csc  
2
( y  0)
( x  0)
( y  0)
Unit Circle
The unit circle is a circle of radius 1
centered at the origin.
Trigonometric Functions
on Unit Circle
Let t be any real number, and let P( x, y ) be the point corresponding to
t when the number line is wrapped onto the unit circle as described above.
Then
sin t  y
cos t  x
tan t 
y
( x  0)
x
1
y
1
sec t 
x
x
cot t 
y
csc t 
( y  0)
( x  0)
( y  0)
The 16-Point Unit Circle
Example Using one Trig
Ratio to Find the Others
Find sin  and cos  , given tan   4 / 3 and cos   0.
Example Using one Trig
Ratio to Find the Others
Find sin  and cos  , given tan   4 / 3 and cos   0.
Since tan  is positive the terminal side is either in QI or QIII.
The added fact that cos is negative means that the terminal
side is in QIII. Draw a reference triangle with r  5, x  -3,
and y  -4.
sin   -4 / 5 and cos  -3/ 5
Periodic Function
A function y  f (t ) is periodic if there is a positive number c such that
f (t  c)  f (t ) for all values of t in the domain of f . The smallest such
number c is called the period of the function.