Post on 14-Jan-2016
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