MATH 5

Yeah, this thing. Honestly, I think we all glanced at it saying, "I'll memorize that one day," but now is the time to ACTUALLY memorize it. This image is just suppose to start you off, the ultimate goal is to hopefully get you to generate it by yourself. 

Note: Although this is named the unit circle, the main take away of this concept is the generation of the first quadrant, Quadrant 1. After the first quadrant is created, the other three quadrants can be generated through the knowledge and repetition of Steps 1 and 2.  

Step one is to understand the basics of the unit circle. Cosine is represented as the "x-value" of an ordered pair, sine is represented as the "y-value" of an ordered pair, and tangent is represented by the quotient of sine by cosine or y/x (see the next post below). Additionally, the unit circle starts at the top right quadrant (Quadrant I) and then going counterclockwise to the bottom right quadrant (Quadrant 4).

Step two is to identify which trigonometric functions is positive. The quick trick for this step is to remember: All Students Take Calculus (ASTC) is meant to help you remember All, Sine, Tangent, and Cosine. Therefore, Quadrant 1: is Sine, Cosine, and Tangent is positive. Quadrant 2:  Sine is positive and Cosine and Tangent is negative. Quadrant 3:  Tangent is positive and Sine and Cosine is negative. Quadrant 4:  Cosine is positive and Sine and Tangent is negative. 

Step three is to draw your standard x-y plane and a circle with a radius of one. Note: in Mathematics the word, "unit" refers to value of one. 

Step four is draw three lines in the first quadrant that mimic that of the drawing to the right. These lines will represent your primary angles: π /6 or 30° , π /4 or 45°, and π /3 or 60°. 

Step five is to draw in five empty coordinate pairs at: 0°, 30°,  45°,  60°,  90°. The coordinate pairs at 0° and 90° will be (1,0) and (0,1) respectively. which then can be rewritten as  (√4/2,√0/2) and (√0/2,√4/2). The remaining pairs just write them in as (√(blank)/2,√(blank)/2). 

Step six starting with the x coordinate first, count down starting from 4 at 0° to 0 at 90°. Therefore starting from 0° your ordered pairs right now are: (√4/2,√0/2), (√3/2,√blank/2), (√2/2,√blank/2), (√1/2,√blank/2), (√0/2,√4/2). 

Step six starting with the y coordinate first, count up starting from 0 at 0° to 4 at 90°. Therefore starting from 0° your ordered pairs right now are: (√4/2,√0/2), (√3/2,√1/2), (√2/2,√2/2), (√1/2,√3/2), (√0/2,√4/2). 

Revised, by Justin, Math Tutor, 9.28.20