Download Trig Functions 4 - Unit Circle

Math 30-1
Trigonometric Functions: Lesson #4
The Unit Circle
Objective: By the end of this lesson, you will be able to:
Is the graph of a circle a function?
This means that we will not be able to express the equation in a simple y  format.
The equation of a circle comes from the Pythagorean Theorem. Consider a circle with its centre
at the origin and a radius of r. Let P x, y  be an arbitrary point on the circle, as shown below.
The radius of the circle can be found using the
Pythagorean Theorem:
• Px, y 
r
This is the equation of a circle with centre (0, 0) and
radius r.
e.g. 1) Write the equation of a circle with centre (0, 0) and radius 7 units.
The unit circle is:
Equation of the Unit Circle:

5 
, y  is on the unit circle. Find all the possible values of y, and explain
e.g. 2) The point  
3


why there is more than one answer.
Math 30-1
Trigonometric Functions: Lesson #4
The unit circle has applications to trigonometry and angles in standard position. Consider an
angle in standard position intersecting the unit circle at a point P  , where  is the measure of
the angle:
P  •

Recall: For any point (x, y) on the terminal
arm of an angle in standard position:
On the unit circle (r = _____) , these ratios
become:
sin  
sin  
cos  
cos  
tan  
tan  
 5 
e.g. 3) Determine the exact coordinates of P  .
 4 
 3 1
,   .
e.g. 4) Identify the measure of the central angle  for 0    2 such that P    
2
2

* Complete the handout “Radians in Standard Position (The Unit Circle)”:
Part 2: Coordinates of the Unit Circle
e.g. 5) What is the length of an arc of the unit circle subtended by a central angle of

?
3
Math 30-1
Trigonometric Functions: Lesson #4
* The length of an arc of the unit circle is
 1 3
 is on the unit circle, determine the following:
e.g. 6) If P      ,

2
2


a) The length of the arc subtended by  .
b) The coordinates of P    .
2

e.g. 7) The point P     x,   lies on the unit circle such that 180    270 .
7

a) Determine the value of x.
b) Determine the value of sec .
c) Determine the value of tan  .
d) Determine the measure of  to the nearest tenth of a degree.
Assignment:
p. 186-189 #1-6, 10, 13, 16
p. 202 #8