Transcript Section 4.2 - Shelton State Community College

Section 4.2
The Unit Circle
About the unit circle
• A unit circle is a circle with a radius of 1 whose
center is at the origin.
• The standard equation of every unit circle is
x  y 1
2
2
A Picture
The Wrapping Function
• In a unit circle, the measure of a central angle
is equal to the length of the intercepted arc.
• It is important to note that this measure is a
real number.
• Let P(s) = (x,y) be a function that has, as its
input, an arc length s measured from (1,0)
and, as its output, the ordered pair on the unit
circle at the other endpoint of s.
Six Trigonometric Functions
•
•
•
•
•
•
Sine: sin s = y
Cosine: cos s = x
Tangent: tan s = y/x
Cosecant: csc s = 1/y
Secant: sec s = 1/x
Cotangent: cot s = x/y
A Picture
Examples—Find the six trig
functions
3 4
P ( s )   , 
5 5
 3 1
P ( s )  
, 
 2 2
P ( s )  0,1
Trig Function Properties: Even and
Odd
•
•
•
•
Cosine and Secant are Even:
cos(-t) = cos t
sec(-t) = sec t
Sine, Cosecant, Tangent, and Cotangent are
Odd:
• sin(-t) = -sin t
csc(-t) = -csc t
• tan(-t) = -tan t
cot(-t) = -cot t
Reciprocal Identities
1
sin t 
csc t
1
csc t 
sin t
1
cos t 
sec t
1
sec t 
cos t
1
tan t 
cot t
1
cot t 
tan t
Quotient Identities
sin t
t ant 
cost
cost
cot t 
sin t
Pythagorean Identities
sin t  cos t  1
2
2
1  t an t  sec t
2
2
1  cot t  csc t
2
2
Periodicity
• Periodicity refers to the tendency of a function
to repeat itself.
• Sine and Cosine have period 2”pi”.
sin(t  2 )  sin t
cos(t  2 )  cost
• Tangent and Cotangent have period “pi”.
tan(t   )  tant
cot(t   )  cot t
Using Your Calculator
• Know how to convert your scientific calculator
from DEGREE mode to RADIAN mode.
• Use your calculator’s reciprocal function to get
values for SEC, CSC, and COT.