Unit 2

Trigonometry

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Part 2.1B corresponds to section 4.2, starting on page 180 of your Pre-Calculus 12 textbook.

Just as a mechanic has wrenches and a tailor has a tape measure, mathematicians also have tools they use regularly. This section explores the unit circle—a tool that will be helpful when defining and determining trigonometric ratios for angles less than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» and greater than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»90«/mn»«mo»§#176;«/mo»«/mstyle»«/math».

The Unit Circle

The unit circle is often drawn on top of angles in standard position. To do this, simply draw a circle with a radius of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», centered at the origin. Here, the word “unit” refers to a single unit, or a radius of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».

This circle will intersect the axes at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math».

Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» be the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»y«/mi»«/mrow»«/mfenced»«/mstyle»«/math», where the terminal arm of an angle in standard position intersects the unit circle.

If a vertical line is drawn from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis, a right triangle is formed with side lengths «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».

From here, it can be seen that the coordinates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» are related by the Pythagorean theorem.


This equation defines the unit circle and can be used to determine if a point lies on the unit circle. It can also be used to determine one coordinate of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«/mstyle»«/math» when the other coordinate is known.