– Angles and the Unit Circle

Angles and the Unit CircleFor each measure, draw an angle with its vertex at the origin of the coordinate plane, use the positive x-axis as one ray of the angle. Do we remember what this is called?

1. 90° 2. 45° 3. 30°

4. 150° 5. 135° 6. 120°

Angles and the Unit Circle

1. 2. 3.

4. 5. 6.

For each measure, draw an angle with its vertex at the origin of the coordinate plane, use the positive x-axis as one ray of the angle. Do we remember what this is called?

1. 90° 2. 45° 3. 30°

4. 150° 5. 135° 6. 120°

Standard Position

The Unit Circle

1-1

1

-1

The Unit Circle-Radius is always one unit-Center is always at the origin

30

Let’s pick a point on the unit circle. The positive angle always goes counter-clockwise from the x-axis.

The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle.

In order to determine the sine and cosine we need a right triangle.

cos30 ,sin 30

The Unit Circle

1-1

1

-1

The angle can also be negative. If the angle is negative, it is drawn clockwise from the x axis.

- 45

Angles and the Unit CircleFind the measure of the angle.

Since 90 + 60 = 150, the measure of the angle is 150°.

The angle measures 60° more than a right angle of 90°.

The angle formed by the terminal side of the angle in standard position and the closest x axis is called the reference angle.

Here the reference angle is 30º

Angles and the Unit CircleSketch each angle in standard position and find the reference angle for each.

a. 48° b. 310° c. –170°

Reference is the same Reference is 50º Reference is 10º

Let’s Try SomeDraw each angle of the unit circle.

a.45o

b.-280 o

c.-560 o

The Unit CircleThe Unit Circle

Definition: A circle centered at the origin with a radius of exactly one unit.

|-------1-------|(0 , 0) (1,0)(-1,0)

(0, 1)

(0, -1)

What are the angle measurements of What are the angle measurements of each of the four angles we just each of the four angles we just found?found?

180°

90°

270°

0°

360°2π

π/2

π

3π/2

0

The Unit Circle

1-1

1

-1

Let’s look at an example

30

The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle.

In order to determine the sine and cosine we need a right triangle.

The Unit Circle

1-1

1

-1

30

Create a right triangle, using the following rules:1.The radius of the circle is the hypotenuse.2.One leg of the triangle MUST be on the x axis. 3.The second leg is parallel to the y axis.

30

601

Remember the ratios of a 30-60-90 triangle-

2

The Unit Circle

1-1

1

-1

30

30

601

2

X- coordinate

Y- coordinate

P

The Unit Circle

1

-1

1

-1

30

60

1

2

The X- coordinate is the horizontal distance of the smaller triangle

The Y- coordinate is the vertical distance of the smaller triangle

P

3

You can see why the x co-orodinate is cosine and the y co-ordinate is sine when we overlap the two triangles to create similar triangles.

1

2

The smaller triangle has a hypotenuse of 1 unit, the radius of the unit circle which is half our identity triangle.

Angles and the Unit CircleFind the cosine and sine of 135°.

Use a 45°-45°-90° triangle to find sin 135°.

From the figure, the x-coordinate of point A

is – , so cos 135° = – , or about –0.71. 22

22

opposite leg = adjacent leg

0.71 Simplify.

= Substitute. 22

The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 13 –0.71 and sin 135° 0.71.

Angles and the Unit CircleFind the exact values of cos (–150°) and sin (–150°).

Step 1: Sketch an angle of –150° in standard position. Sketch a unit circle.

x-coordinate = cos (–150°)y-coordinate = sin (–150°)

Step 2: Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.)

Angles and the Unit Circle(continued)

The triangle contains angles of 30°, 60°, and 90°.

Step 3: Find the length of each side of the triangle.

hypotenuse = 1 The hypotenuse is a radius of the unit circle.

shorter leg = The shorter leg is half the hypotenuse.12

12

32longer leg = 3 = The longer leg is 3 times the shorter leg.

32

12

Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so

cos (–150°) = – , and sin (–150°) = – .

Let’s Try SomeDraw each Unit Circle. Then find the cosine and sine of each angle.

a.45o

b.120o

45° Reference Angles - 45° Reference Angles - CoordinatesCoordinatesRemember that the unit circle is overlayed on a coordinate plane (that’s how

we got the original coordinates for the 90°, 180°, etc.)

Use the side lengths we labeled on the QI triangle to determine coordinates.

45°135°

315°

225°

( , )

( , )

( , )

( , ) 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

π/4

3π/4

5π/4 7π/4

30-60-90 Green Triangle30-60-90 Green TriangleHolding the triangle with the single fold down and double fold to the left, label each

side on the triangle.

Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the blue butterfly.

60° Reference Angles - 60° Reference Angles - CoordinatesCoordinates

Use the side lengths we labeled on the QI triangle to determine coordinates.

60°120°

300°

240°

( , )

( , )

( , )

( , ) 2

3

2

3

2

1

2

3

2

1

2

3

2

1

2

1

2

1

2

3

π/32π/3

4π/3 5π/3

30-60-90 Yellow Triangle30-60-90 Yellow TriangleHolding the triangle with the single fold down and double fold to the left, label each

side on the triangle.

Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the green butterfly.

30° Reference Angles 30° Reference Angles We know that the quadrant one angle formed by the triangle is 30°.

That means each other triangle is showing a reference angle of 30°. What about in radians?

Label the remaining three angles.

30°150°

330°210

°

π/6

7π/6

5π/6

11π/6

30° Reference Angles - 30° Reference Angles - CoordinatesCoordinatesUse the side lengths we labeled on the QI triangle to determine coordinates.

30°150°

330°210

°

( , )

( , )

( , )

( , ) 2

1

2

1

2

3

2

1

2

3

2

1

2

3

2

3

2

3

2

1

π/6

7π/6

5π/6

11π/6

Final ProductFinal Product

The Unit CircleThe Unit Circle