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Trigonometric Functions

The Unit Circle
The Unit Circle

Definition: A circle whose center is the origin and whose radius
has a length of one.
Based on the definition, give the coordinates for the x- and yintercepts for the diagram below.
(__,
0 , __)
1
1 , __)
0
(__,
(__,
-1 , __)
0
(__,
0 , __)
-1
The Unit Circle

Let’s determine the coordinates on the unit circle for a 45 angle.
2
2
2 __)
2
(__,
1
45
2
2
2
2
What is the x-coordinate of
the point? 22
2
The y-coordinate? 2
Drop the perpendicular from the point
on the circle to the positive x-axis.
What type of triangle is created?
An isosceles right triangle
What is the length of the hypotenuse? Why?
1 because it is a unit circle
What is the relationship between the
hypotenuse and a leg of an isosceles
right triangle?
hl 2
What is the length of each leg? l 
1
2

2
2
The Unit Circle
What would be the coordinates
of the point on the circle if we
draw a 225 angle? Explain.
What would be the coordinates
of the point on the circle if we
draw a 135 angle? Explain.

2
2
2
2
(__, __)
135
225

2
2

2
2
(__, __)
The Unit Circle
The same procedure can be used to find
the coordinates for a 30 angle and a
60 angle. Visiting all quadrants would
result in the following figure.
What would be the coordinates
of the point on the circle if we
draw a 315 angle? Explain.
315
2
2

2
2
(__, __)
The Trigonometric Functions

Let t be the measure of a central angle and let (x, y) be the point on the unit
circle corresponding to t. The following are the definitions for the six
trigonometric functions based on the unit circle.
sin t  y
1
csc t 
y
cost  x
1
sec t 
x
y
tan t 
x
x
cot t 
y
The Trigonometric Functions
How to determine the trigonometric values for a given angle
Ex. 1: Evaluate the six trigonometric functions for t 

6
Step 1: Identify the quadrant the terminal side of the angle

is located.
Quadrant I
Step 2: Identify the coordinates that correspond with that angle.

 3 1
, 

2
2

Step 3: Follow the definitions
1
sin t  y 
2
y
tan t  
x
3
cos t  x 
2
1
csc t   2
y
1
2
3
2
1
3


3
3
1
2 2 3
sec t  

x
3
3
x
cot t  
y
3
1
2
2
 3
The Trigonometric Functions
5
Ex. 2: Evaluate the six trigonometric functions for
4
Step 1: Identify the quadrant the terminal side of the angle
is located.
t
Quadrant II
Step 2: Identify the coordinates that correspond with that angle.

2
2

,


2 
 2
Step 3: Follow the definitions
2
sin t  y 
2
2
cos t  x  
2
2
y
tan t   22  1
x  2
1
2
csc t  
 2
y
2
1
2
sec t   
 2
x
2
x  22
cot t   2  1
y
2
The Trigonometric Functions
Ex. 2: Evaluate the six trigonometric functions for
t
7
3
In this problem the value of the angle exceeds 2. Find the coterminal angle
whose value lies between 0 and 2 7
7  6 
3
 2 
3

3
Step 1: Identify the quadrant the terminal side of the angle is located.
Quadrant I
Step 2: Identify the coordinates that correspond with that angle.
1 3
 ,

2
2


Step 3: Follow the definitions
3
sin t  y 
2
cos t  x 
1
2
y
tan t   3
x
csc t 
1 2 3

y
3
sec t 
1
2
x
cot t 
x
3

y
3
The Trigonometric Functions

Recapping for evaluating the six trigonometric
functions for any given angle.




If the given angle does not lie between 0 and 2, find its
simplest positve coterminal angle and use that value.
Identify the quadrant the terminal side for the given angle lies.
Determine the coordinates on the unit circle for the given
angle.
Follow the definitions for the six trigonometric functions.
Even and Odd Trig Functions

What is an even function?

An even function is when one substitutes a negative value into
the original function and the outcome is the same as its
positive value.
f (- x) = f (x)
Ex. f (x) = x2
f (2) = 22 = 4
f (-2) = (-2)2 =4
Therefore, f (-2) = f (2) and f (x) = x2 is an even
function.
Even and Odd Trig Functions

What is an odd function?

An odd function is when one substitutes a negative value into the
original function and the outcome is the opposite of its positive value.
f (- x) = - f (x)
Ex. f (x) = x3
f (2) = 23 = 8
f (-2) = (-2)3 = - 8
Therefore, f (-2) = - f (2) and f (x) = x3 is an odd function.
Even and Odd Functions

Let’s determine whether or not the sine function is even or odd.
 f(x) = sin x



What is the sin 30? ½
What is the sin (- 30)? sin (- 30) = - sin (30)
sin (- 30) = -( ½ ) = - ½
½
Because the sin (- 30) = - sin (30), the sine function is odd.
sin (-x) = - sin x
Even and Odd Trig Functions

Lets determine whether the cosine function is even or odd.
 f (x) = cos x



What is the cos 60? ½
What is the cos (- 60)? ½
cos (- 60) = cos 60
cos (- 60) = ½
Because the cos (- 60) equals cos 60, cosine is an even function
cos (- x) = cos x
Even and Odd Trig Functions

Determine what the other 4 trig functions are – even or odd

tan (x) 
odd function

csc (x) 
odd function

sec (x) 
even function

cot (x) 
odd function
Evaluating Trig Fuctions
with a Calculator

Steps

Set the calculator into the correct mode

Casio
 From the MENU select RUN
 Press SHIFT, then MENU (above MENU you should see SET
UP)
 Scroll down and highlight Angle
 Select F1 for Deg, Select F2 for Rad
 Press EXE (Blue key at bottom of calculator)
 Type the measure of the angle into the calculator
 Press EXE
Evaluating Trig Fuctions
with a Calculator

Using the calculator find the values for
the following (round to 4 decimal places):




1.
2.
3.
4.
sin 214 -0.5592
tan (-175 ) 0.0875
cos 5/9 -0.1736
sin 14/5 0.5878
Evaluating Trig Fuctions
with a Calculator

To evaluate cosecant, secant, or cotangent on the
calculator, follow these steps:

Casio




Make sure you are in the right mode (degree or radian)
In parentheses type the trig function on the calculator that is the
reciprocal function of the one being evaluated.
On the outside of the parentheses, press SHIFT, then ). A - 1 should
appear next to the parentheses.
Example: Evaluate csc 40




Make sure you are in the degree mode.
Type into the calculator (sin 40)
Press SHIFT, then ). Your screen should now read (sin 40) - 1
Press EXE. Your answer should be 1.5557
Evaluating Trig Fuctions
with a Calculator

Using the calculator find the values for
the following (round to 4 decimal places):




1.
2.
3.
4.
sec 297 2.2027
cot 19/12 -0.2679
csc (- 11.78) 1.4128
sec /2 Ma ERROR -
WHY?
4.2 Trigonometric Fuctions:
The Unit Circle

Can you
 Sketch the unit circle and place key angles and
coordinates on it?
 Explain how these coordinates are derived using
either 30, 45 , or 60 ?
 Determine the trig value for certain angles using the
unit circle?
 Explain why a trig function is either even or odd?
 Evaluate a trig function using the calculator?