Deriving the Unit Circle
Inquiry Activity Summary:
1.
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| Before Reduction (http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_30_60_90.gif) |
A 30° Triangle has three sides to it: adjacent (x), opposite (y), and hypotenuse (r). The opposite side is always x. The adjacent will be xradical3 and the hypotenuse will be 2x. To derive the unit circle from this triangle, the hypotenuse must equal 1. To get that to happen, all three sides have to be divided by 2x and simplified. After reduction the values become 1 for hypotenuse, radical3/2 for adjacent, and 1/2 for opposite. The resulting sides can be plotted and the coordinate near the 60° measure will be the coordinate used in the unit circle. The pair of coordinates can be used for the 30° angle measure and all its supplemental angles (150°, 210°, 330°). For the 30° degree triangle we know the coordinate is (radical3, 1/2). An example of all the steps being taken can be found in the bottom:
2.
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| Triangle before reduction (http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_45_45_90.gif) |
A 45° Triangle is composed of two sides that are the same length (x and y) and a hypotenuse. The equal sides are labeled as x and the hypotenuse is xradical 2. To derive the unit circle values, we must get the hypotenuse to equal 1. By dividing every value by xradical2 we can accomplish this and simplify to get our distance values that can be plotted out. The final values after reducing are 1 for hypotenuse, and radical2/2 for both adjacent and opposite. The resulting coordinate near the 45° angle measure will be used for 45° in the circle and all its supplemental angles (135°, 225°, 315°). That value is (xradical2, xradical2). An example of this derivation can be seen in the following image:
3.
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| Triangle before reduction (http://img.sparknotes.com/figures/B/b21b8be4e663f46072a3c3b19b657891/306090.gif) |
A 60° Triangle has the same sides as a 30° triangle. The opposite side is always xradical3. The adjacent will be x and the hypotenuse will be 2x. Since the location of the degrees change, the same simplifications are taken but they end up in different values. The final measures after reducing are 1 for hypotenuse, 1/2 for adjacent, and radical3/2 for opposite. The coordinate pair near the 30 measure will be used in the unit circle, the pair is (1/2, radical3/2). This value is the same for 60° and all its supplementary angles (120°, 240°, 300°). An example can be found in this image:
4.
This activity helps me derive the Unit Circle because the triangles reflect in the circle throughout all four quadrants. If reflected upon the y-axis, the coordinates measure the same from the bottom up. If reflected from there down the x-axis, the coordinates measure the same from the top down. If shifted from the first quadrant down across the x-axis, the coordinates measure the same from the top down. The only things that change are the signs and those sign changes can are explained in section 5.
5.The triangle in this activity lies in quadrant I since both x y values in the (x,y) coordinates are positive. If you change quadrants, the coordinate values would change. If you drew the triangles in quadrant II, the x value would become negative and the y would remain positive. If you drew the triangles in quadrant III both the x and y values would become negative. If you drew the triangles in quadrant IV, the x value would remain positive and the y values would change to negative. To view an example of different quadrants and the change in coordinates that occur along with them view this image derived by myself from repeating triangles:
Inquiry Activity Reflection:
1. The coolest thing I learned from this activity was how easy it is to follow the triangles throughout the different quadrants.
2. This activity will help me in this unit because when I feel lost if I forget my coordinates or angle measures, I can always use the first five to figure out the rest of the unit circle or try to derive the triangles again if my mind if blank.
3. Something I never realized before about special right triangles and the unit circle is that the values in the unit circle correlate with the values found when the special right triangles' hypotenuse is reduced to one.
References:
30 degree triangle image: http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_30_60_90.gif
60 degree triangle image:http://img.sparknotes.com/figures/B/b21b8be4e663f46072a3c3b19b657891/306090.gif
45 degree triangle image:http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_45_45_90.gif
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