Unit 2

Trigonometry

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Part 2.1A corresponds to section 4.1, starting on page 166 of your Pre-Calculus 12 textbook.

Radians

Most measurements have several units associated with them. For example, length can be measured in metres, feet, and miles; volume can be measured in litres, gallons, and cubic centimetres; and time can be measured in seconds, days, and years. Similarly, angles can be measured in degrees, radians, sextants, arc minutes, and arc seconds. The focus of this section will be the angular unit of radians.

In the Investigation, you attempted to determine the measure of an angle subtended by an arc length of one radius. Were you able to determine the angle to be approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»57«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»3«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math»?

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»«mo»§#960;«/mo»«mi»r«/mi»«/mrow»«/mstyle»«/math», so if one radius of the circumference is used to make an angle, that angle will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»r«/mi»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«mi»r«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» of the entire circle. A complete circle is «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¨»«mrow»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfrac»«mo»§#8226;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»§#8784;«/mo»«mn»57«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»3«/mn»«mo»§#176;«/mo»«/mrow»«/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»«mo»§#960;«/mo»«/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»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» radians. (If this statement seems unclear, compare it to “If one centimetre is one hundredth of a metre, then a metre must contain «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»100«/mn»«/mstyle»«/math» centimetres”.) This means «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»radians«/mi»«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¨»«mrow»«mo»§#960;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»radians«/mi»«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, it simplifies a lot of advanced math involving angles.

Note: Generally, when radians are used, no unit is shown. It is assumed an angle measure of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«/mstyle»«/math» is in radians, while «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#176;«/mo»«/mstyle»«/math» is in degrees.