1. Copyright © 2003, N. Ahbel What is a radian and why do we use them?* * (Why not just use degrees?)

2. Copyright © 2003, N. Ahbel What is a radian? When the length of the arc equals the length of the radius The angle has a measure of 1 radian A radian is an angle measurement that gives the ratio:  = 1 radian

3. Copyright © 2003, N. Ahbel What is a radian? the length of the radius The angle has a measure of 2 radians A radian is an angle measurement that gives the ratio: When the length of the arc is twice as long as  = 2 radians

4. Copyright © 2003, N. Ahbel What is a radian? the length of the radius The angle has a measure of 3 radians A radian is an angle measurement that gives the ratio: When the length of the arc is three times as long as  = 3 radians

5. Copyright © 2003, N. Ahbel What is a radian? the length of the radius The angle has a measure of π radians A radian is an angle measurement that gives the ratio: When the length of the arc is π times as long as  = π radians

6. Copyright © 2003, N. Ahbel What is a radian? A radian is an angle measurement that gives the ratio: radius=1 A “unit circle” is a circle with radius = 1 and center at the origin. (0,0) (1,0)  The radian measure of an angle for a unit circle is equal to the length of the arc that it subtends.

7. Copyright © 2003, N. Ahbel Why do we use radians? Using radians make some formulas simple while the same formula is more complicated with degrees: Using radians Angle measurement, , is Using degrees Angle measurement, , is where s is the arc length and r is the radius

8. Copyright © 2003, N. Ahbel Why do we use radians? Radians are very important in Calculus. The area between y = sin (x) and the x-axis from 0 < x < 180 is approximately 114.6 when graphed in degrees.

9. Copyright © 2003, N. Ahbel Why do we use radians? Radians are very important in Calculus. The area between y = sin (x) and the x-axis from 0 < x < π is exactly 2 when graphed in radians.

10. Copyright © 2003, N. Ahbel First usages of radians and degrees Muir, a mathematician, and James T. Thomson, a physicist, were working independently during the late nineteenth century to develop a new unit of angle measurement. They met and agreed on the name radian, a shortened form of the phrase radial angle. Different names were used for the new unit until about 1900. Today the term radian is in common usage." Source: Algebra 2, Bettye C. Hall and Mona Fabricant"James, 1991 Prentice-Hall "The yonge sonne That in the Ram is foure degrees vp ronne" (OED2). He again used the word in about 1391 in A Treatise on the Astrolabe: "9. Next this folewith the cercle of the daies, that ben figured in manere of degres, that contenen in nombre 365, dividid also with longe strikes fro 5 to 5, and the nombre in augrym writen under that cercle." Source: Chaucer's Canterbury Tales, 1386

11. Copyright © 2003, N. Ahbel What is a radian and why do we use them?* * (Why not just use degrees?)