Coordinates of the Point Where the Terminal Side of an Angle with a Measure of 5

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. Circle((0, 0), 1) → c
3. LaTeX("\text{0 radians}") → zero
4. LaTeX("\text{Sketch the angle.}") → text3
5. If(angleNum / angleDen < 1, Sequence[i / angleDen, i, 1, angleDen - 1], Sequence[1 + i / angleDen, i, 1, angleDen - 1]) → list1
6. Zip((cos((a * π)), sin((a * π))), a, list1) → list2
7. Zip(Segment[(0, 0), a], a, list2) → list3
8. Segment((0, 0), C) → a
9. Segment((0, 0), (1, 0)) → b
10. Angle(A, (0, 0), C) → θ
11. 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
12. 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 π radians to find the }\cr\text{measure of the reference angle.}"], If[1 < factor ≤ 1.5, LaTeX["\text{Subtract π radians 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 2π radians to find the }\cr\text{measure of the reference angle.}"]]]) → subtraction
13. If(0 < factor ≤ 0.5, angleNum, If[0.5 < factor ≤ 1, angleDen - angleNum, If[1 < factor ≤ 1.5, angleNum - angleDen, (2 * angleDen) - angleNum]]) → refNum
14. 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}
15. LaTeX((((("\theta=\frac{" * angleNum) * "\pi}{") * angleDen) * "} radians")) → Problem
16. LaTeX((((((("\textcolor{#CC0000}{\sin(\theta_{ref})=\sin\left(\frac{" * refNum) * "}{") * angleDen) * "}\pi\right)\approx") * sinRef) * "}")) → text11
17. LaTeX((((((("\textcolor{#CC0000}{\cos(\theta_{ref})=\cos\left(\frac{" * refNum) * "}{") * angleDen) * "}\pi\right)\approx") * cosRef) * "}")) → text12
18. LaTeX("\text{Determine the coordinates of point }\textcolor{#006633}{C}.") → text14
19. If(0 < factor ≤ 0.5, LaTeX["\text{(+,+)}"], If[0.5 < factor ≤ 1, LaTeX["\text{(-,+)}"], If[1 < factor ≤ 1.5, LaTeX["\text{(-,-)}"], LaTeX["\text{(+,-)}"]]]) → signs
20. 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
21. LaTeX("\text{Move the slider to solve the problem.}") → text0
22. LaTeX("\text{Sketch a unit circle, and then label π radians and 2π radians.}") → text1
23. LaTeX("\text{A half rotation (180°) is π radians and a full rotation (360°) is 2π radians. }\cr\text{Notice that 0 radians and 2π radians are in the same location on the unit circle, }\cr\text{but represent different angle measures.}") → text2
24. If(angleNum / angleDen < 1, LaTeX[(((((((((((((((((("\frac{" * angleNum) * " π}{") * angleDen) * "}\text{is the same as }\frac{") * angleNum) * "}{") * angleDen) * "} π. \cr\text{In other words, the terminal side is }\frac{") * angleNum) * "}{") * angleDen) * "}\text{ of the way from 0 to π.}\cr\text{Thus, imagine the semicircle between 0 radians and π radians split into ") * angleDen) * " parts }\cr\text{and then sketch the angle }\frac{") * angleNum) * "}{") * angleDen) * "}\text{ of the way around the semicircle.}")], LaTeX[(((((((((((((((((("\frac{" * angleNum) * " π}{") * angleDen) * "}\text{is the same as }\left(1+\frac{") * newNum) * "}{") * angleDen) * "}\right) π. \cr\text{In other words, the terminal side is }\frac{") * newNum) * "}{") * angleDen) * "} \text{ of the way from π to 2π. }\cr\text{Thus, imagine the semicircle between π radians and 2π radians split into ") * angleDen) * " parts }\cr\text{and then sketch the angle }\frac{") * newNum) * "}{") * angleDen) * "}\text{ of the way around the semicircle.}")]) → text4
25. 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
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 }\frac{\pi}{2} \text{radians, the sine}\cr\text{and cosine of the reference angle are always non-negative.}") → text13
28. 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{ are } ") * signs) * ".")) → text15
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{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
31. LaTeX((((("\text{The terminal side falls in Quadrant } " * quadrant) * ".\cr") * subtraction) * "")) → text7
32. If(0 < factor ≤ 0.5, LaTeX[(((("\textcolor{#CC0000}{frac{" * refNum) * "}{") * angleDen) * "}π \text{ radian(s)}}")], If[0.5 < factor ≤ 1, LaTeX[(((((((((((("\textcolor{#CC0000}{\frac{" * angleDen) * "}{") * angleDen) * "}π -\frac{") * angleNum) * "}{") * angleDen) * "}π =\frac{") * refNum) * "}{") * angleDen) * "}π \text{ radian(s)}}")], If[1 < factor ≤ 1.5, LaTeX[(((((((((((("\textcolor{#CC0000}{\frac{" * angleNum) * "}{") * angleDen) * "}π -\frac{") * angleDen) * "}{") * angleDen) * "}π =\frac{") * refNum) * "}{") * angleDen) * "}π \text{ radian(s)}}")], LaTeX[(((((((((((("\textcolor{#CC0000}{\frac{2(" * angleDen) * ")}{") * angleDen) * "}π -\frac{") * angleNum) * "}{") * angleDen) * "}π =\frac{") * refNum) * "}{") * angleDen) * "}π\text{ radian(s)}}")]]]) → text8
33. LaTeX((((((((("\text{The reference angle for }\frac{" * angleNum) * "\text{ }\pi}{") * angleDen) * "}\text{ radians is }\frac{") * refNum) * "\text{ }\pi}{") * angleDen) * "}\text{ radian(s).}")) → text9
34. LaTeX((("\text{The coordinates of point }\textcolor{#006633}{C}\text{ are approximately } \textcolor{#006633}{" * C) * "}.")) → text17