Unit 4A

Trigonometry Part 1

Lesson 1: Trigonometry Review

The unit circle shown was not complete as it did not contain the radian measurements for each angle. Before the angles can be labelled in radians, radians must be reviewed.

Radians

The circumference of a circle is related to the radius by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»C«/mi»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«mi»r«/mi»«/mrow»«/mstyle»«/math», so if one radius length of the circumference (arc length of one radius) is subtended by a central angle, that central angle will measure «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»r«/mi»«mi»C«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mi»r«/mi»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«mi»r«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» of the entire circle. A complete circle measures «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math», so one radian is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»§#8784;«/mo»«mn»57«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#176;«/mo»«/mstyle»«/math».


If one radian is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfrac»«/mstyle»«/math» of a circle, then a circle must contain «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mstyle»«/math» radians. This means «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mstyle»«/math» radians«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math», or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/mstyle»«/math» radians «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mn»180«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

The radian may seem like an unusual unit, but because it is derived from the circle itself, instead of splitting a circle into an arbitrary «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»360«/mn»«/mstyle»«/math» units, the radian simplifies a great deal of advanced math involving angles.