Finding Six Trigonometric Functions Using Unit Circle

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Corner(1) → A
4. Corner(3) → B
5. Element(fractions, n) → frac
6. FractionText(frac) → fractext
7. Circle((0, 0), 1) → c
8. Segment((0, 0), C) → a
9. Segment((0, 0), (1, 0)) → b
10. Angle(b, a) → α
11. Segment((0, 0), (-1, 0)) → d
12. If(0 < frac ≤ 0.5, Angle[b, a], If[0.5 < frac ≤ 1, Angle[a, d], If[1 < frac < 1.5, Angle[d, a], Angle[a, b]]]) → β
13. Element(coords, n) → showcoord
14. Segment(C, (x(C), 0)) → e
15. Segment((0, 0), (x(C), 0)) → f
16. Midpoint(e) → D
17. Midpoint(f) → E
18. LaTeX(Element[xvalue, n]) → xlabel
19. LaTeX(Element[yvalue, n]) → ylabel1
20. Midpoint(a) → G
21. Segment((0, 0), F) → g
22. Segment((0, 0), (x(F), 0)) → h
23. Segment((x(F), 0), F) → i
24. LaTeX(Element[yvalue, n]) → ylabel2
25. Midpoint(g) → H
26. Midpoint(h) → I
27. Midpoint(i) → J
28. LaTeX(Element[xvalue, n]) → xlabel2
29. Element(refractions, n) → refrac
30. FractionText(refrac) → refractext
31. LaTeX((("\text{Substitute }\textcolor{#000099}{" * fractext) * "\pi}\text{ for }\textcolor{#000099}{\theta}.")) → text5
32. LaTeX((((((("\textcolor{#000099}{\tan\left(" * fractext) * "\pi\right)=\frac{\sin\left(") * fractext) * "\pi\right)}{\cos\left(") * fractext) * "\pi\right)}}")) → text6
33. LaTeX((((((("\textcolor{#000099}{\tan\left(" * fractext) * "\pi\right)=\frac{") * ylabel1) * "}{") * xlabel) * "}}")) → text8
34. LaTeX("\text{Simplify.}") → text9
35. LaTeX("\textcolor{#000099}{\sec(\theta)=\frac{1}{\cos(\theta)}}") → text12
36. LaTeX((((((((("\textcolor{#000099}{\sec\left(" * fractext) * "\pi\right)=\frac{1}{\cos\left(") * fractext) * "\pi\right)}=\frac{1}{") * xlabel) * "}=") * Element[secval, n]) * "}")) → text13
37. LaTeX("\textcolor{#000099}{\csc(\theta)=\frac{1}{\sin(\theta)}}") → text14
38. LaTeX((((((((("\textcolor{#000099}{\csc\left(" * fractext) * "\pi\right)=\frac{1}{\sin\left(") * fractext) * "\pi\right)}=\frac{1}{") * ylabel1) * "}=") * Element[cscval, n]) * "}")) → text15
39. LaTeX("\textcolor{#000099}{\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}}") → text16
40. LaTeX((((((((((((("\textcolor{#000099}{\cot\left(" * fractext) * "\pi\right)=\frac{\cos\left(") * fractext) * "\pi\right)}{\sin\left(") * fractext) * "\pi\right)}=\frac{") * xlabel) * "}{") * ylabel1) * "}=") * Element[cotval, n]) * "}")) → text17
41. LaTeX((((("\text{Use the unit circle to find the values of }\textcolor{#000099}{\cos\left(" * fractext) * "\pi\right)}\text{ and }\textcolor{#000099}{\sin\left(") * fractext) * "\pi\right)}.")) → text1
42. LaTeX((((((((((((("\text{The coordinates of the point }\textcolor{#000099}{\theta=" * fractext) * "\pi}\text{ on the unit circle are }\cr\textcolor{#000099}{") * showcoord) * "}\text{ which correspond to the trigonometric ratios }\cr\textcolor{#000099}{\cos\left(") * fractext) * "\pi\right)=") * xlabel) * "}\text{ and }\textcolor{#000099}{\sin\left(") * fractext) * "\pi\right)=") * ylabel1) * "}.")) → text3
43. LaTeX((("\text{Use the ratio identity }\textcolor{#000099}{\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}}\text{ to find the value of }\textcolor{#000099}{\tan\left(" * fractext) * "\pi\right)}.")) → text4
44. LaTeX((((((((("\text{Substitute }\textcolor{#000099}{" * xlabel) * "}\text{ for }\textcolor{#000099}{\cos\left(") * fractext) * "\pi\right)}\text{ and }\cr\textcolor{#000099}{") * ylabel1) * "}\text{ for }\textcolor{#000099}{\sin\left(") * fractext) * "\pi\right)}.")) → text7
45. LaTeX("\text{Use the reciprocal identities to find the remaining trigonometric values.}") → text11
46. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
47. LaTeX((("\textcolor{#000099}{\theta=" * fractext) * "\pi}")) → Problem
48. LaTeX((((("\textcolor{#000099}{\tan\left(" * fractext) * "\pi\right)=") * Element[tanval, n]) * "}")) → text10
49. If(1 ≤ n ≤ 5, LaTeX[(((("\text{Since }\textcolor{#000099}{0 \le " * fractext) * "\pi \le \frac{\pi}{2}}\text{ (or, in degrees, }\textcolor{#000099}{0\le ") * degrees) * "^{\circ} \le 90^{\circ}}\text{),}\cr\text{the angle is in the first quadrant. The angle is its own}\cr\text{reference angle.}")], If[6 ≤ n ≤ 9, LaTeX[(((((("\text{Since }\textcolor{#000099}{\frac{\pi}{2} < " * fractext) * "\pi \le \pi }\text{ (or, in degrees, } \textcolor{#000099}{ 90^{\circ} < ") * degrees) * "^{\circ} \le 180^{\circ}}\text{),}\cr\text{the angle is in the second quadrant. The reference angle is }\textcolor{#006633}{") * refractext) * "\pi}\cr\text{measured up from the negative }\textit{x}\text{-axis as shown.}")], If[10 ≤ n ≤ 13, LaTeX[(((((("\text{Since }\textcolor{#000099}{ \pi < " * fractext) * "\pi \le \frac{3\pi}{2} }\text{ (or, in degrees, } \textcolor{#000099}{ 180^{\circ} < ") * degrees) * "^{\circ} \le 270^{\circ}}\text{),}\cr\text{the angle is in the third quadrant. The reference angle is }\textcolor{#006633}{") * refractext) * "\pi}\cr\text{measured down from the negative }\textit{x}\text{-axis as shown.}")], LaTeX[(((((("\text{Since }\textcolor{#000099}{ \frac{3\pi}{2} < " * fractext) * "\pi \le 2\pi }\text{ (or, in degrees, }\textcolor{#000099}{ 270^{\circ} < ") * degrees) * "^{\circ} \le 360^{\circ}}\text{),}\cr\text{the angle is in the fourth quadrant. The reference angle is }\textcolor{#006633}{") * refractext) * "\pi}\cr\text{measured down from the positive }\textit{x}\text{-axis as shown.}")]]]) → text2