Transcript Thinking Mathematically by Robert Blitzer

Trigonometric
Functions: The Unit
Circle
1.2
Objectives
 Students will be able to identify a unit circle
and describe its relationship to real
numbers.
 Students will be able to use a unit circle to
evaluate trigonometric functions.
 Students will be able to use the domain and
period to evaluate sine and cosine
functions.
 Students will be able to use a calculator to
evaluate trigonometric functions.
The Unit Circle
(0, 1)
(-1, 0)
(1, 0)
(0, -1)
Definitions of the Trigonometric
Functions in Terms of a Unit Circle
If t is a real number and (x, y) is a
point on the unit circle that
corresponds to t, then
sin t  y
cos t  x
1
csc t  , y  0
y
1
sec t  ,x  0
x
y
tan t  , x  0
x
x
cot t  , y  0
y
Use the Figure to find the values of the
trigonometric functions at t=/2.
Solution:
(0,1)
The point P on the unit circle that
Corresponds to t= /2 has coordinates
(0,1). We use x=0 and y=1 to find the
Values of the trigonometric functions


sin
csc
2

2
 y 1

tan t 
cos
2
x0
1 1
 x 0
  1 cot    0
y 1
2 y 1
y
1 1
and sec t    undefined
x
x 0
/2
/2
(1,0)
2
2
x  y 1
THE DOMAIN AND RANGE OF THE SINE AND COSINE
FUNCTIONS AND THEIR PERIOD
The domain of the sine function and the cosine
function is the set of all real numbers
 The range of these functions is the set of all real
numbers from -1 to 1, inclusive.
 The period is 2π.
 This means it repeats every
 Periodic: f(t+c)=f(t) where c= 2π.
 Page 152 # 36, 42

Definition of a Periodic Function
A function f is periodic if there exists a
positive number p such that
f(t + p) = f(t)
For all t in the domain of f. The smallest
number p for which f is periodic is called
the period of f.
Periodic Properties of the Sine
and Cosine Functions
sin(t + 2) = sin t and cos(t + 2) = cos t
The sine and cosine functions are periodic
functions and have period 2.
Periodic Properties of the
Tangent and Cotangent Functions
tan(t + ) = tan t and cot(t + ) = cot t
The tangent and cotangent functions are
periodic functions and have period .
Even and Odd Trigonometric
Functions
The cosine and secant functions are even.
cos(-t) = cos t
sec(-t) = sec t
The sine, cosecant, tangent, and
cotangent functions are odd.
sin(-t) = -sin t
csc(-t) = -csc t
tan(-t) = -tan t
cot(-t) = -cot t
EXAMPLE

Use the value of the trigonometric function at t =
/4 to find sin (- /4 ) and cos(- /4 ).
sin
csc

4

4
2
2
cos
 2
sec


4

4

2
2
 2
tan
cot

4

4
1
1
Solution:
sin(-t) = -sin t, so sin(- /4 ) = -sin(/4 ) = -2/2
cos(-t) = cos t, so cos(- /4 ) = cos(/4 ) = 2/2
Try it: Pg. 151 # 38, 40, 46, 48, 50, 56
Homework: # 5 – 57 odd
Right Angle Trigonometry
Objectives
 Students will be able to evaluate
trigonometric functions of acute angles.
 Students will be able to use fundamental
trigonometric identities.
 Students will be able to use a calculator
to evaluate trigonometric functions.
 Students will be able to use
trigonometric functions to model and
solve real life problems.
The Six Trigonometric Functions
The figure below shows a right triangle with one of its acute
angles labeled . The side opposite the right angle is known as the
hypotenuse. The other sides of triangle are described by the position
relative to the acute angle . One side is opposite  and one is adjacent
to .
Hypotenuse
Side opposite .

Side adjacent to .
RIGHT TRIANGLE DEFINITIONS OF
TRIGONOMETRIC FUNCTIONS
sin  
Opp
Hyp
csc  
Hyp
Opp
cos  
Adj
Hyp
sec  
Hyp
Adj
tan  
Opp
Adj
cot  
Adj
Opp
Hyp
Opp

Adj
How does compare to the unit circle?
 Page 160 #8, 12

45°
30°
√2/2
1
2
2
√3
45°
60°
1
1
1
Reciprocal Identities
1
sin t 
csct
1
csc t 
sin t
1
cos t 
sect
1
sec t 
cos t
1
tan t 
cot t
1
cot t 
tan t
Quotient Identities
sin t
tan t 
cos t
cos t
cot t 
sin t
PYTHAGOREAN IDENTITIES
2
2
sin t  cos t  1
2
2
1  tan t  sec t
1  cot t  csc t
2
2
COFUNCTION IDENTITIES
sin( 90   )  cos 
cos(90   )  sin 
tan( 90   )  cot 
cot (90   )  tan 
sec( 90   )  csc 
csc(90   )  sec 
o
o
o
o
o
o
Examples
• Page 161 #32, 38, 44, 46, 58, 62, 66, 70
• Homework: 5 – 47 odd, 57 – 67 odd