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WARM UP
A student calculated the following. What are the miscalculations
he made?
40
a) Find sin θ
sin θ = 30/50
50
30
θ
b) Find sin θ given the point P(5, –3) is on the
terminal side of angle θ
a2 + b 2 = c 2
52 + –32 = c2
25 – 9 = c2
16 = c2
c=4
sin θ = –3/4
c) cos (5π/4) = (5 x 3.14)/4 = 3.925
What you’ll learn about
• The Unit Circle
• The 16-Point Unit Circle
• Trigonometric Functions of Real Numbers
… and why
• Extending trigonometric functions beyond
triangle ratios opens up a new world of
applications
Trigonometric Functions of any Angle
• What is sin θ in terms of x, y, r?
• What is cos θ in terms of x, y, r?
• What is tan θ in terms of x, y, r?
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.
P(x, y)
Trigonometric Functions of Real Numbers
Instead of angles, we now consider the
trigonometric functions of real numbers
Let t be any real number, and let P( x, y ) be the point corresponding to
the angle of t radians. Then, since r = 1 in the Unit Circle:
1
sin t  y
csc t 
( y  0)
y
1
cos t  x
sec t 
( x  0)
x
y
x
tan t 
( x  0)
cot t 
( y  0)
x
y
Slide 4- 6
Unit Circle
The real number t always corresponds to the
point P(cos t, sin t) on the Unit Circle
t is expressed in radians
P(cos t, sin t)
First Quadrant
degrees
0˚
30˚
45˚
60˚
90˚
radians
0 rad
x = cos t
To convert degrees to radians, multiply by π/180
y = sin t
Second Quadrant
degrees
120˚
135˚
150˚
180˚
radians
x = cos t
π rad
To convert degrees to radians, multiply by π/180
y = sin t
Third Quadrant
degrees
210˚
225˚
240˚
270˚
radians
x = cos t
To convert degrees to radians, multiply by π/180
y = sin t
Fourth Quadrant
degrees
300˚
315˚
330˚
360˚
radians
x = cos t
2π rad
To convert degrees to radians, multiply by π/180
y = sin t
HOMEWORK
Unit Circle Worksheet
EXIT TICKET
What did you learn today?