The Unit Circle

Part II

(With Trig!!)

MSpencer

Multiples of 90°, 2

0°, 0 360°, 2180°,

90°, 2

270°, 3 2

The Quadrants (with Angles)

0°, 0 360°, 2180°,

90°, 2

270°, 3 2

Q I0° < < 90°

0 < < 2

QII90° < < 180°

< < 2

QIII180° < < 270°

< < 32

QIV270° < < 360°

< < 232

The Unit Circle

r = 1

Remember it is called a unit circle because the radius is one unit.

So let’s add in ordered pairs to the unit circle.

Multiples of 90°, 2

0°, 0180°,

90°, 2

270°, 3 2

(1, 0)

(0, 1)

(1, 0)

(0, 1)

r = 1r

=

1

r = 1

r = 1

45°, 4

45°, 4

45°

45°Notice that 45° or forms one of the two special right triangles from geometry.

4

45°, 4

45°

45°

Let’s review this triangle from geometry.

Opposite the congruent, 45° angles are congruent sides.

These sides are the legs of the right triangle. So the triangle is an isosceles right triangle.

45°, 4

45°

45°

Let’s call the two congruent legs s.

s

s

2s

22

The hypotenuse is the length of either leg, s, times ; thus, s .

45°, 4

45°

45°

Lastly, now remember that the hypotenuse is the radius of the unit circle, which means it must equal one.

Solve for s.

s

s

2 1s

2 1s 1

2s 2

2 2

2

45°, 4

45°, 4

45°

45°

2

2

2

2

1 The distance across the bottom side of the triangle represents the x-coordinate while the right, vertical side represent y.

2 2,

2 2

Signs and Quadrants

0°, 0180°,

90°, 2

270°, 3 2

Q I(+, +)

The signs of each ordered pair follow the signs of x and y for each quadrant.

Q II(, +)

Q III(, )

Q IV(+, )

Multiples of 45°, 4

135°, 3 4

315°, 7 4

45°, 4

225°, 5 4

2 2,

2 2

2 2,

2 2

2 2,

2 2

2 2,

2 2

45°

45°

2

2

2

2

2

2

2

2

45°

45°

45°

45°2

2

2

2

60°, 3

60°, 3

Notice that 60° or forms the other special right triangle from geometry.

3

60°

30°

60°, 3

Let’s review this triangle from geometry.

Call the the smallest side opposite 30° s.

60°

30°

The hypotenuse is twice the smallest side, or 2s.

3s3

The medium side opposite 60° is times the smallest side, or .

s

2s 3s

2 1s 1

2s

60°

30°

s

2s = 1 3s

60°, 3

The hypotenuse is the radius of the unit circle, which means it must equal one.

Solve for s.

3

2

The medium side opposite 60° is

60°, 3

60°, 3 1 3

,2 2

Notice that since the triangle is taller than it is wide, that the y-coordinate is larger than the x-coordinate.

1

2

3

2

y

x

1 3,

2 2

1

2

3

2

Multiples of 60°, 3

120°, 2 3

300°, 5 3

60°, 3

240°, 4 3

1

2

3

2

1 3,

2 2

1 3,

2 2

1 3,

2 2

30°,

30°, 6

Notice this is the same special right triangle as for 60° except the x and y coordinates are switched.

6

y

x

60°30°

1

2

3

2

3 1,

2 2

Multiples of 30°, 6

150°, 5 6

330°, 116

30°, 6

210°, 7 6

60°30°

1

2

3

2

3 1,

2 2

3 1,

2 2

3 1,

2 2

3 1,

2 2

Ordered Pairs and Trig

From geometry, recall SOHCAHTOA, which defines sine, cosine, and tangent.

sine (Sin) =

cosine(Cos) =

tangent (Tan) =

pposite

ypote

O

H nuse

djacent

ypote

A

H nuse

O

A

pposite

djacent

30°, 6

60°30°

1

2

3

2

3 1,

2 2

Ordered Pairs and Trig

Cos 30° = 3

djacent 2ypotenuse 1

A

H

3

2cos 30° =

Notice that the cosine of the angle is simply the x-coordinate!

30°, 6

60°30°

1

2

3

2

3 1,

2 2

Ordered Pairs and Trig

Sin 30° = 1pposite 2

ypotenuse 1

O

H

1

2sin 30° =

Notice that the sine of the angle is simply the y-coordinate!

And this is true for ANY angle, often called .

cos = x

sin = y

Ordered Pairs: Cosine & Sine

(x, y) (cos , sin )

Signs for Cosine and Sine

0°, 0180°,

90°, 2

270°, 3 2

Q I(+, +)

The “signs” of cosine and “sine” follow the signs of x and y in each quadrant.

Q II(, +)

Q III(, )

Q IV(+, )

So in QII, for instance, cosine is negative while sine is positive.

The Whole Unit Circle Together (Grouped)

0°, 0 (1, 0)

90°, 2

(0, 1)

180°, (1, 0)

270°, 3 2

(0, 1)

45°, 4 2 2

,2 2

135°, 3 4

2 2,

2 2

225°, 5 42 2

,2 2

315°, 7 4 2 2

,2 2

60°, 3 1 3

,2 2

120°, 2 31 3

,2 2

240°, 4 31 3

,2 2

300°, 5 3 1 3

,2 2

30°, 6 3 1,2 2

150°, 5 6

3 1,2 2

210°, 7 63 1,2 2

330°, 11

6 3 1,2 2

The Whole Unit Circle Together (In Ascending Order)

0°, 0 (1, 0)

90°, 2

(0, 1)

180°, (1, 0)

270°, 3 2

(0, 1)

45°, 4 2 2

,2 2

135°, 3 4

2 2,

2 2

225°, 5 42 2

,2 2

315°, 7 4 2 2

,2 2

60°, 3 1 3

,2 2

120°, 2 31 3

,2 2

240°, 4 31 3

,2 2

300°, 5 3 1 3

,2 2

30°, 6 3 1,2 2

150°, 5 6

3 1,2 2

210°, 7 63 1,2 2

330°, 11

6 3 1,2 2