Our goal in todays lesson will be to build

the parts of this unit circle. You will then want to get it memorized

because you will use many facts from this

to answer other Pre Calc and Calculus

questions

2

3,

2

1

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)

Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value?

(1,0)

(0,1)

(0,-1)

(-1,0)

x = 1/2

You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y.

122 yx

12

1 22

y

4

32 y

2

3y

2

3,

2

1

2

3,

2

1

We can also use special right triangles and trig functions to find important points on the unit circle!

30

60

45

45

2x

3x

x

x

x

x2

θ

opposite = y

adjacent = x

hypotenuse

= 1

yy

hyp

opp

1sin

x

y

adj

opptan

xx

hyp

adj

1)cos(

3 Trig Functions you learned in Geometry

In the unit circle, the hypotenuse will be the radius of the circle; therefore, it will be 1 !

x

1

cos

1sec

y

x

tan

1cot

y

1

sin

1)csc(

3 More Trig Functions based on reciprocals in the unit circle

We will look closer at these later in the lesson!

From last slideCos(ɵ) = xSin(ɵ) = yTan(ɵ)= y/x

Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle!

30

60

30

60

X

X2

3X

2

3

1

2

1

2

3

1 0

0 1

-1 0

0 -1

30 : 60 : 90

x : : 2x3x

1 : : 2

1

Radius = 1

30

60

30

601

2

1

2

3

There is a relationship between the coordinates of a point P on the circle and the sine and cosine of the angle (θ) containing P

P ( cos θ, sin θ )

30sin

30cos2

3x

2

1y

1 0

0 1

-1 0

0 -1

From earlier slideCos(ɵ) = xSin(ɵ) = y

Now lets complete quadrant I for 45° and 60° as well!

Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle!

45

45

45

45

Radius=1

2

2

2

1

1

2X

2

2

2

2

1 0

0 1

-1 0

0 -1

45 : 45 : 90

x : x :

: : 12

22

2

Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle!

45

45

45

451

2

2

2

2

P ( cos θ, sin θ )

45sin

45cos2

2x

2

2y

1 0

0 1

-1 0

0 -1

Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle!

60

3060

30

12

2

30 : 60 : 90

x : : 2x3x

12

1

2

3 : :

60sin

60cos

2

3

2

1

2

1x

2

3y

Same result as when we used circle formula!!!

1 0

0 1

-1 0

0 -1

Here is the unit circle divided into 8 pieces.

45°

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.

45° is the reference angle for 135°, 225°, and 315 °

Can you figure out how many degrees are in each division?

Reference Angles45° is the reference angle for 135°, 225°, and 315 °

30° is the reference angle for 150°, 210°, and 330 °

60° is the reference angle for 120°, 240°, and 300 °

Use the points we found in quadrant I and consider the signs of each quadrant and the reference angles to find the

remaining coordinates in the unit circle.

Complete the angles from the reference angle: 30°

30°

30°

2

1,

2

3

90°

0°

120°

180°

210°

270°

330°

60°150°

240°300°

2

1,

2

3

2

1,

2

3

2

1,

2

3

Complete the angles from the reference angle: 60°

30°

30°

90°

0°

120°

180°

210°

270°

330°

60°150°

240°300°

2

3,

2

1

2

3,

2

1

2

3,

2

1

2

3,

2

1

Can you figure out what these angles would be in radians?

The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4.

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

4

7sin

2

2

0,1

4

2

4

3

4

5

2

34

7

Can you figure out what the angles would be in radians?

30°

It is still halfway around the circle and the upper half is divided into 6 pieces so each piece is /6.

30°

2

1,

2

3

90°

1,0

0°

120°

180° 0,1

210°

270°

330°

1,0

0,1

60°150°

240°300°

2

3,

2

1

2

3,

2

1

2

3,

2

1

2

1,

2

3

2

1,

2

3

2

1,

2

3

2

3,

2

1

We'll see

them all put

together on the

unit circle on the next screen.

6

3

2

3

2

6

5

6

7

3

4

2

33

5 6

11 2

You should memorize

this. This is a great

reference because you can

figure out the trig

functions of all these angles quickly.

2

3,

2

1

2

3,

2

1

Look at the unit circle and determine sin 420°.

All the way around is 360° so we’ll need more than that. We see that it will be the same as sin 60° since they are coterminal angles. So sin 420° = sin 60°.

sin 420° =

2

3

2

3sin 780° =

How about finding values other than just sine and cosine?

x

1

cos

1sec

y

x

tan

1cot

y

1

sin

1)csc(

3 More Trig Functions based on reciprocals in the unit circle

From last slideCos(ɵ) = xSin(ɵ) = yTan(ɵ)= y/x

Add these to your notes

Sec ( ) =

Csc (45°) =

Tan (690°) =

Cot

3

4

3

3

3

3

2

1