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

122 yx

So points on this circle must satisfy this equation.

(1,0)

(0,1)

(0,-1)

(-1,0)

(1,0)

(0,1)

(0,-1)

(-1,0)

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.

2

1

2

31

sin

cos2

1

121

Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

tan 3

2123

2

3,2

12

3

123

Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division?

45°

We can label this all the way around 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.

45°

2

2,

2

290°

1,0

0°

135°

2

2,

2

2

180° 0,1

225°

270°315°

2

2,

2

2

2

2,

2

2

1,0

225sin2

2

0,1

These are easy to

memorize since they

all have the same value

with different

signs depending

on the quadrant.

• Use the unit circle to find the two values.

)90sec()180cos( oo and

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 Q3. If sin( ) = find the values of the remaining 5 trig functions.

5

4