1. Trigonometric Functions
The Unit Circle

2. 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 y-
intercepts for the diagram below.
(__, __)
0 , 1
1 , 0
-1 , 0
0 , -1

3. The Unit Circle Drop the perpendicular from the point
Let’s determine the coordinates on the unit circle for a 45 angle.
Drop the perpendicular from the point
on the circle to the positive x-axis.
(__, __)
What type of triangle is created? 1
An isosceles right triangle
45
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?
What is the x-coordinate of
the point?
What is the length of each leg?
The y-coordinate?

4. The Unit Circle (__, __) (__, __) What would be the coordinates
of the point on the circle if we
draw a 135 angle? Explain.
What would be the coordinates
of the point on the circle if we
draw a 225 angle? Explain.
135
(__, __)
225
(__, __)

5. 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
(__, __)

6. 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.

7. The Trigonometric Functions
How to determine the trigonometric values for a given angle
Ex. 1: Evaluate the six trigonometric functions for
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.
Step 3: Follow the definitions

8. The Trigonometric Functions
Ex. 2: Evaluate the six trigonometric functions for
Step 1: Identify the quadrant the terminal side of the angle
is located.
Quadrant II
Step 2: Identify the coordinates that correspond with that angle.
Step 3: Follow the definitions

9. The Trigonometric Functions
Ex. 2: Evaluate the six trigonometric functions for
In this problem the value of the angle exceeds 2. Find the coterminal angle
whose value lies between 0 and 2
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.
Step 3: Follow the definitions

10. 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.

11. 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.

12. 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.

13. Even and Odd Functions ½ - ½
Let’s determine whether or not the sine function is even or odd.
f(x) = sin x
What is the sin 30?
What is the sin (- 30)?
sin (- 30) = - sin (30)
sin (- 30) = -( ½ ) = - ½
Because the sin (- 30) = - sin (30), the sine function is odd.
sin (-x) = - sin x
- ½

14. Even and Odd Trig Functions
Lets determine whether the cosine function is even or odd.
f (x) = cos x
What is the cos 60?
What is the cos (- 60)?
cos (- 60) = cos 60
cos (- 60) = ½
Because the cos (- 60) equals cos 60, cosine is an even function
cos (- x) = cos x

15. Even and Odd Trig Functions
Determine what the other 4 trig functions are – even or odd
tan (x) 
csc (x) 
sec (x) 
cot (x) 
odd function
odd function
even function
odd function

16. 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

17. Evaluating Trig Fuctions with a Calculator
Using the calculator find the values for the following (round to 4 decimal places):
1. sin 214
2. tan (-175 )
3. cos 5/9
4. sin 14/5
0.0875
0.5878

18. 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

19. Evaluating Trig Fuctions with a Calculator
Using the calculator find the values for the following (round to 4 decimal places):
1. sec 297
2. cot 19/12
3. csc ( )
4. sec /2
2.2027
1.4128
Ma ERROR - WHY?

20. 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?