CCGPS AA 5.1.3 Example 1: Coordinates of Intersection of 330° Terminal Side with

GeoGebra Commands & Parameters

1. GCD(angleDeg, 180) → common
2. LaTeX(steps) → counter
3. Circle((0, 0), 1) → c
4. If(factor < 1, angleDeg / common, (angleDeg - 180) / common) → newNum
5. LaTeX("\text{Sketch the angle.}") → text3
6. If(factor < 1, Sequence[(i * 180 / newDen), i, 1, newDen - 1], Sequence[180 + (i * 180 / newDen), i, 1, newDen - 1]) → list1
7. Zip((cos((a * π / 180)), sin((a * π / 180))), a, list1) → list2
8. Zip(Segment[(0, 0), a], a, list2) → list3
9. Segment((0, 0), C) → a
10. Segment((0, 0), (1, 0)) → b
11. Angle(A, (0, 0), C) → θ
12. If(0 < factor ≤ 0.5, LaTeX["\text{I}"], If[0.5 < factor ≤ 1, LaTeX["\text{II}"], If[1 < factor ≤ 1.5, LaTeX["\text{III}"], LaTeX["\text{IV}"]]]) → quadrant
13. If(0 < factor ≤ 0.5, LaTeX["\text{The measure of the original angle is the measure of the reference angle.}"], If[0.5 < factor ≤ 1, LaTeX["\text{Subtract the measure of the original angle from 180° to find the }\cr\text{measure of the reference angle.}"], If[1 < factor ≤ 1.5, LaTeX["\text{Subtract 180° from the original angle measure to find the}\cr\text{measure of the reference angle.}"], LaTeX["\text{Subtract the measure of the original angle from 360° to find the }\cr\text{measure of the reference angle.}"]]]) → subtraction
14. If(0 < factor ≤ 0.5, angleDeg, If[0.5 < factor ≤ 1, 180 - angleDeg, If[1 < factor ≤ 1.5, angleDeg - 180, 360 - angleDeg]]) → refDeg
15. If(0 < factor ≤ 0.5, Angle[A, (0, 0), C], If[0.5 < factor ≤ 1, Angle[C, (0, 0), B], If[1 < factor ≤ 1.5, Angle[B, (0, 0), C], Angle[C, (0, 0), A]]]) → θ_{ref}
16. LaTeX("\text{Determine the coordinates of point }\textcolor{#006633}{C}.") → text14
17. If(0 < factor ≤ 0.5, LaTeX["\text{(+,+)}"], If[0.5 < factor ≤ 1, LaTeX["\text{(-,+)}"], If[1 < factor ≤ 1.5, LaTeX["\text{(-,-)}"], LaTeX["\text{(+,-)}"]]]) → signs
18. If(0 < factor ≤ 0.5, LaTeX["\textcolor{#006633}{(\cos(\theta),\sin(\theta))}=\textcolor{#CC0000}{(\cos(\theta_{ref}),\sin(\theta_{ref}))}"], If[0.5 < factor ≤ 1, LaTeX["\textcolor{#006633}{(\cos(\theta),\sin(\theta))}=\textcolor{#CC0000}{(-\cos(\theta_{ref}),\sin(\theta_{ref}))}"], If[1 < factor ≤ 1.5, LaTeX["\textcolor{#006633}{(\cos(\theta),\sin(\theta))}=\textcolor{#CC0000}{(-\cos(\theta_{ref}),-\sin(\theta_{ref}))}"], LaTeX["\textcolor{#006633}{(\cos(\theta),\sin(\theta))}=\textcolor{#CC0000}{(\cos(\theta_{ref}),-\sin(\theta_{ref}))}"]]]) → text16
19. LaTeX((("\theta=" * angleDeg) * "^{\circ}")) → Problem
20. LaTeX((((("\textcolor{#CC0000}{\cos(\theta_{ref})=\cos\left(" * refDeg) * "^{\circ}\right)\approx") * cosRef) * "}")) → text12
21. LaTeX((((("\textcolor{#CC0000}{\sin(\theta_{ref})=\sin\left(" * refDeg) * "^{\circ}\right)\approx") * sinRef) * "}")) → text11
22. LaTeX("\text{Move the slider to solve the problem.}") → text0
23. LaTeX("\text{A half rotation (π radians) is 180° and a full rotation (2π radians) is 360°. }\cr\text{Notice that 0° and 360° are in the same location on the unit circle, }\cr\text{but represent different angle measures.}") → text2
24. If(angleDeg / 180 < 1, LaTeX[(((((((((((((("\text{The terminal side of }" * angleDeg) * "°\text{ is }\frac{") * angleDeg) * "}{180}=\frac{") * newNum) * "}{") * newDen) * "}\text{ of the way from 0° to 180°.}\cr\text{Thus, imagine the semicircle between 0° and 180° split into ") * newDen) * " parts and then}\cr\text{sketch the angle }\frac{") * newNum) * "}{") * newDen) * "}\text{ of the way around the semicircle.}")], LaTeX[(((((((((((((("\text{The terminal side of }" * angleDeg) * "°\text{ is }\frac{") * angleDeg) * "-180}{180}=\frac{") * newNum) * "}{") * newDen) * "}\text{ of the way from 180° to 360°. }\cr\text{Thus, imagine the semicircle between 180° and 360° split into ") * newDen) * " parts and then }\cr\text{sketch the angle }\frac{") * newNum) * "}{") * newDen) * "}\text{ of the way around the semicircle.}")]) → text4
25. LaTeX((((("\text{The reference angle for }" * angleDeg) * "°\text{ is }") * refDeg) * "°.")) → text9
26. LaTeX("\text{Determine }\textcolor{#CC0000}{\sin(\theta_{ref})}\text{ and }\textcolor{#CC0000}{\cos(\theta_{ref})}\text{ in order to determine }\textcolor{#006633}{\sin(\theta)}\text{ and }\textcolor{#006633}{\cos(\theta)}.") → text10
27. LaTeX("\text{Because the reference angle is always between 0° and 90°, the sine and cosine of the}\cr\text{reference angle are always non-negative.}") → text13
28. LaTeX((("\text{The coordinates of point }\textcolor{#006633}{C}\text{ are approximately } \textcolor{#006633}{" * C) * "}.")) → text17
29. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text18
30. LaTeX("\text{0°") → zero
31. LaTeX("\text{Sketch a unit circle, and then label 0°, 180°, and 360°.}") → text1
32. LaTeX("\text{Determine the measure of the reference angle, which is the measure of the}\cr\text{angle between the terminal side and the x-axis. }") → text6
33. LaTeX((((("\text{The terminal side falls in Quadrant } " * quadrant) * ".\cr") * subtraction) * "")) → text7
34. LaTeX("\text{The coordinates of point }\textcolor{#006633}{C}\text{ are given by } \textcolor{#006633}{(\cos(\theta),\sin(\theta))}\text{, which can be evaluated }\cr\text{from the reference angle }\textcolor{#CC0000}{\theta_{ref}}.") → text5
35. LaTeX((((("\text{Since point }\textcolor{#006633}{C}\text{ and the terminal side lie in Quadrant }" * quadrant) * ", \cr\text{the signs of the coordinates of point }\textcolor{#006633}{C}\text{ follow the pattern } ") * signs) * ".")) → text15
36. If(0 < factor ≤ 0.5, LaTeX[(("\textcolor{#CC0000}{" * angleDeg) * "°}")], If[0.5 < factor ≤ 1, LaTeX[(((("\textcolor{#CC0000}{180°-" * angleDeg) * "°=") * refDeg) * "°}")], If[1 < factor ≤ 1.5, LaTeX[(((("\textcolor{#CC0000}{ " * angleDeg) * "°-180°=") * refDeg) * "°}")], LaTeX[(((("\textcolor{#CC0000}{ 360°-" * angleDeg) * "°=") * refDeg) * "°}")]]]) → text8