Transcript 5.3 - Collinsville, Oklahoma

5.3
The Unit Circle
A circle with center at (0, 0) and radius 1 is called a unit circle.
The equation of this circle would be
x  y 1
2
2
(0,1)
(-1,0)
(1,0)
(0,-1)
So points on this circle must satisfy this equation.
We know all of the sides of this triangle. The bottom leg is just
the x value of the point, the other leg is just the y value and
the hypotenuse is always 1 because it is a radius of the circle.
(0,1)
1
(-1,0)
1 3
 ,

2 2 


3
2

sin  
(1,0)
1
2
tan  
(0,-1)
cos 
3
2  3
1
2
1
21

1 2
3
2  3
1
2
Notice the sine is just the y value of the unit circle point and
the cosine is just the x value.
Here is the unit circle divided into 8 pieces. Can you figure
out how many degrees are in each division?
These are
0,1
easy to

 2 2
2 2
90°
memorize




,
 2 2 
 2 , 2 

 135°


since they
45°
all have the
2
sin 225  
same value
2
with
45°
1,0 180°
0° 1,0 different
signs
depending
225°
on the

2
2
315°


2
2




,
 2 , 2 
quadrant.
270°



2
2 


We can label this all the way around
0,1 with how many degrees an
angle would be and the point on the unit circle that corresponds
with the terminal side of the angle. We could then find any of the
trig functions.
• Use the unit circle to
find the two values.
cos(180 )andsec(90 )
o
o
Example:
• Use the unit circle to find the values of the
six trigonometric functions for a 135
degree angle.
We actually don’t even need points
to be on the units circle…
• For example: Find the values of the six
trigonometric functions for angle theta in
standard position if a point with
coordinates (5, -12) lies on its terminal
side.
For any angle “A” in standard
position…
•
•
•
•
•
•
•
•
•
•
•
The sine function: sin A = y / r
The cosine function: cos A = x / r
The tangent function: tan A = y / x = sin A / cos A
The cosecant function: csc A = r / y = 1 / sin A
The secant function: sec A = r / x = 1 / cos A
The cotangent function: cot A = x / y = 1 / tan A = A = cos
A / sin A
Suppose  is an angle in standard
position whose terminal side lies in
4
Q3.
If sin(  ) = 5
find the values of the remaining 5 trig
functions.