Unit 2

Trigonometry

Arc Length

In the Investigation, you may have noticed that an arc and the angle subtended by that arc use the same amount of the circle. So, if the angle is a quarter of a rotation, the arc will be a quarter of the circumference, and if the angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»48«/mn»«mi mathvariant=¨normal¨»%«/mi»«/mrow»«/mstyle»«/math» of a rotation, the arc length will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»48«/mn»«mi mathvariant=¨normal¨»%«/mi»«/mrow»«/mstyle»«/math» of the circumference. You may even have used this type of thinking to determine the size of a radian.

When using radians, this relationship simplifies. There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» radians in a rotation and there are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» radii in the circumference. So, for each radian of rotation, there is one radius of arc length. If the angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», the arc length will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» radius. If the angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», the arc length will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» radii, and so on.