Radians and Degrees

What the heck is a radian?

• The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.

• Say what now?!

• Radian: the angle made by taking the radius and wrapping it along the edge of a circle.

Radians as Ratios

• Radian measure of an angle, therefore, is the ratio of the angle’s arc length compared to the radius of the circle

• Radians in a full circle:

Radians and Degrees

• We know that there are 2π radians in circle

• We also know that there are 360° in a circle

Converting: Radians to Degrees

• We know so if we want to convert radians to degrees, we can use a proportion.

To solve for degrees, just multiply the radian measure by .

• Ex: Convert radians to degrees.

Practice Converting Radians to Degrees

2. 4

3.

4.

Converting: Degrees to Radians

• To convert degrees to radians we rearrange the proportion we used before to get

• Now to find the radian amount, we just multiply the degree measure by

• Ex: Convert 225° to radians.

Practice Converting Degrees to Radians

1. 320°

2. 45 °

3. 165°

4. 540 °

180 °=𝜋360 °=2𝜋0 °=0

90 °=𝜋2

270 °=3𝜋  2  

45 °=𝜋4

135 °=3𝜋4

225 °=5𝜋4

315 °=7𝜋4

Common Angles In Radians

180 °=𝜋360 °=2𝜋0 °=0

90 °=𝜋2

270 °=3𝜋  2  

60 °=𝜋3

120 °=2𝜋3

240 °=4𝜋3

300 °=5𝜋3

30 °=𝜋6150 °=

5𝜋6

210 °=7𝜋6

330 °=11𝜋6

Common Angles In Radians

180 °=𝜋360 °=2𝜋0 °=0

90 °=𝜋2

270 °=3𝜋  2  

60 °=𝜋3

120 °=2𝜋3

240 °=4𝜋3

300 °=5𝜋3

30 °=𝜋6150 °=

5𝜋6

210 °=7𝜋6

330 °=11𝜋6

45 °=𝜋4

135 °=3𝜋4

225 °=5𝜋4

315 °=7𝜋4

The Unit Circle

Measuring Angles: Terms

• Greek letters are used to name angles. Ex: Alpha ( ) Beta ( ) and Theta ( )

• Initial Side – this is the starting position of the angle

• Terminal Side – this is the ending position of the angle

• Standard Position – vertex on the origin, initial side on the positive x-axis

Measuring Angles: Terms

• Positive Angles – angles that are rotated counterclockwise (CCW)

• Negative Angles – angles that are rotated clockwise (CW)

• Coterminal Angles – angles that have the same initial and terminal sides

Negative Angle

Positive AngleCoterminal Angles

Sketching Angles

• Use the unit circle angles to help you approximate where your angle should be.

• Start in standard position. If the angle is positive, move counterclockwise. If the angle is negative, move clockwise.

• Use arrows to indicate how many (if any) full revolutions the angle has made

Examples:

Finding Coterminal Angles

In Degrees

• To find positive coterminal angle, add where n is any positive integer.

• To find a negative coterminal angle, subtract until the resulting angle is negative.

In Radians

• To find positive coterminal angle, add where n is any positive integer.

• To find a negative coterminal angle, subtract until the resulting angle is negative.

Quadrants and Quadrantal Angles

• The x and y axes split the coordinate plane into 4 quadrants.

• Angles whose terminal side falls on either of the axes are called quadrantal angles• Since the axes fall every quarter

turn, or radians, the quadrantal angles are any multiple of , or

Quadrant 1Quadrant 2

Quadrant 3 Quadrant 4

20

2

2

3 2

2

3