2. What is a Radian? Radian is the ratio between the length of an arc and its radius in a given circle. Rad𝑖𝑎𝑛=𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡h𝑅𝑎𝑑𝑖𝑢𝑠 arc length 1 radian radius radius = arc length
3. Radians Compared to Degrees Radians The radian is a fixed measure of a circle. There are 2𝜋 radians in a circle. Degrees There are 360° (degrees) in a circle. There are 360 equal parts of a circle. This is an arbitrary number; makes working with circle measurements easy.
4. Converting Radians and Degrees Radians -> Degrees 𝑑𝑒𝑔𝑟𝑒𝑒=𝑟𝑎𝑑𝑖𝑎𝑛∙180°𝜋 Example Angle in radians: 𝜋4 𝜋4=𝜋4∙180°𝜋=180°4=45° Degrees -> Radians 𝑟𝑎𝑑𝑖𝑎𝑛=𝑑𝑒𝑔𝑟𝑒𝑒∙𝜋180° Example Angle in degrees: 45° 45°=45°∙𝜋180°=45𝜋180=𝜋4 =0.785
Radian is the ratio between the length of an arc and its radius in a given circle. This means, the measure in radians is the arc length divided by the radius. A measure of one radian occurs when the arc length and the radius have the same measure.
Radians and degrees are two ways to calculate the measure of circles. Radians are a fixed measure of a circle: arc length divided by the radius. There are two pi radians in a circle. On the other hand, there are 360 degrees in a circle. This means that there are 360 equal parts of a circle. 360 is an arbitrary number. It is useful when working with circles because it makes circle measurements easy.
It is important to know how to convert measurements between radians and degrees. When converting from radians to degrees, multiply the radian by 180 divided by pi. When converting from degrees to radians multiply the degree by pi divided by 180. The two examples show how easy it is to convert from radians to degrees and back to radians. If we have an angle of pi fourths radians, multiply by 180 divided by pi. The two pi’s cancel out, and after simplifying, the result is 45 degrees. If we start with 45 degrees, multiply by pi divided by 180. After simplifying, we get pi fourths.
Why do we know to multiply by pi over 180 or 180 over pi? Here is the derivation. The circumference of a circle is equal to 2 pi times the radius. The circumference, or 2 pi r, is analogous to 360 degrees because they are both one full loop of the circle. To get the radian measure of any full circle, we take the definition of a radian and divide 2 pi r by r, the radius. That leaves us with 2 pi radians. Hence 2 pi equals 360 degrees. By isolating the radian on one side, we get 360 over 2 pi. Simplifying gives us 180 over pi. One radian is equal to 180 over pi. To convert between radians and degrees, we use this ratio of 180 over pi or pi over 180. This picture shows various angles and their measures in both degrees and radians.
The information presented in this slide show came from wikipedia.org. The final picture came from the math site at rice university.
