Download 4-3: TRIGONOMETRIC FUNCTIONS ON THE UNIT CIRCLE ANGLES AND THE UNIT CIRCLE

4-3: TRIGONOMETRIC FUNCTIONS
ON THE UNIT CIRCLE
Precalculus
Mr. Gallo
ANGLES AND THE UNIT CIRCLE
Trig functions of angles are used to describe
coordinates
y
1
To find the coordinates of P:
1.Draw height of triangle back to x-axis
(creates reference angle, )
2.Hypotenuse has length of 1 unit, height =
y and base = x
,
3.Use trig functions to find x and y
y
x
sin  
cos  
x
1
1
sin   y
cos   x
4.Rewrite the coordinates of P as:
cos , sin
1
FINDING COSINE AND SINE OF QUADRANTAL ANGLES
Quadrantal Angles
 Terminal side of angle falls on the x- or y-axis
 
 3 
 0  0  , 90   , 180   , 270   , 360  2 
2
 2 
Angles
Coordinates
 0,1
0
 1, 0 
1, 0 
90
180
 0, 1
270
0

2

3
2
360 2
1, 0 
 0,1
 1, 0 
 0, 1
1, 0 
FINDING EXACT VALUES OF COSINE AND SINE
Use special right triangles and trig functions to calculate
the coordinates.
To find the coordinates of P:
1.Draw height of triangle back to x-axis
(creates reference angle, 30°)
y
2.Hypotenuse has length of 1 unit, height =
y and base = x
,
3.Use trig functions to find x and y
1
1
3
1
2
x
30°
cos 30  2
3
sin 30  2
1
1
2
3
1
cos 30 
sin 30 
2
2
4.Rewrite the coordinates of P as:
31
,
2 2
2
FINDING EXACT VALUES OF COSINE AND SINE
Use special right triangles and trig functions to calculate
the coordinates.
To find the coordinates of P:
1.Draw height of triangle back to x-axis
(creates reference angle, 60°)
2.Hypotenuse has length of 1 unit, height =
y and base = x
3.Use trig functions to find x and y
3
1
x
sin 60  2
cos 60  2
1
1
y
,
1
60°
1
2
3
2
3
1
sin 60 
2
2
4.Rewrite the coordinates of P as:
1 3
,
2 2
cos 60 
FINDING EXACT VALUES OF COSINE AND SINE
Use special right triangles and trig functions to calculate
the coordinates.
y
,
1
45°
2
2
2
2
To find the coordinates of P:
1.Draw height of triangle back to x-axis
(creates reference angle, 45°)
2.Hypotenuse has length of 1 unit, height =
y and base = x
3.Use trig functions to find x and y
2
2
x
sin 45  2
cos 45  2
1
1
2
2
sin 45 
cos 45 
2
2
4.Rewrite the coordinates of P as:
2 2
,
2 2
3
REFERENCE ANGLES
Can be used to find the values of any angle .
 Always drawn back to -axis
What are the coordinates of a 135° angle?
1. Create a reference angle between
the terminal side of the angle and
the -axis
2.
180° 135° 45° so the
coordinates are the same as the
triangle formed by a 45°angle.
3.
y
,
1
2
2
,
135°
x
2
2
Complete Guided Practice 3A-3C p.244

3A.
3B. 60
3C. 30
4
SIGNS OF TRIGONOMETRIC FUNCTIONS
y
• Look at the sign of the and
coordinates in each quadrant
• Remember
and
 ,  
 ,  
 ,  
 ,  
x
Quad I Quad II Quad III Quad IV
sin
cos








4
COMPLETE UNIT CIRCLE
Use reference angles to complete the other quadrants.
FINDING TANGENT
Remember:
tan  
opp
y
sin 


adj
x
cos 
y
,
1
45°
x
What is the tan 45° ?
2
sin 45
2 2 
tan 45 
 2 
1
cos 45
2  2 
2
2
5
SIGNS OF TRIGONOMETRIC FUNCTIONS
y
• Look at the sign of the and
coordinates in each quadrant
• Remember
 ,  
 ,  
 ,  
 ,  
x
Quad I Quad II Quad III Quad IV
sin
cos
tan












Homework: p.251 #9, 10, 13-31 odd, 41
6
SIX TRIGONOMETRIC FUNCTIONS ON THE UNIT CIRCLE
Sine, Cosine and Tangent can be used to find the
reciprocal trigonometric functions:
Find the following:
cot 210
3
cot 210  2  3
1

2

7
4
7
1

 2
sec
4
2
2
sec
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
Apply concept of unit circle to finding coordinates of
circle of any size radius:
7
EVALUATING TRIGONOMETRIC FUNCTIONS GIVEN A POINT
Let 4,3 be a point on the terminal side of an angle
in standard position. Find the exact values of the six
trigonometric functions of
r  x2  y2 
 4 
2
 32  25  5
sin  
y 3

r 5
cos  
x 4

r
5
tan  
y 3

x 4
csc  
r 5

y 3
sec  
r
5

x 4
cot  
x 4

y 3
Complete Guided Practice 1A-1B p.242
3
4
3
1
 5
2 5
1A. sin =
cos =
tan =
tan =
1B. sin =
cos =
5
5
4
2
5
5
5
5
4
5
csc =
sec =
cot =
cot =2
csc = 5 sec =
3
4
3
2
Homework: p.251 #1-7 odd, 11, 12-16 even, 33-39 odd,
43-57odd
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