飞机飞行轨迹与角度三角函数示例

GeoGebra 命令及参数

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Corner(1) → A
4. Corner(3) → B
5. FractionText(Element[fractions, n]) → fractext
6. If((n < 4) ∨ (9 < n), "east", "west") → direction
7. Ray((0, 0), (cos(angleR), sin(angleR))) → a
8. Ray((0, 0), (1, 0)) → b
9. Angle(b, a) → θ
10. If((n < 4) ∨ (9 < n), Vector[(0, 0), (dist, 0)], Vector[(0, 0), ((-dist), 0)]) → u
11. If((n < 4) ∨ (9 < n), (dist / 2, 0), (((-dist)) / 2, 0)) → C
12. LaTeX(dist) → distlab
13. If(n ≤ 3, Ray[(dist, 0), Vector[(0, 1)]], If[4 ≤ n ≤ 6, Ray[((-dist), 0), Vector[(0, 1)]], If[7 ≤ n ≤ 9, Ray[((-dist), 0), Vector[(0, -1)]], Ray[(dist, 0), Vector[(0, -1)]]]]) → c
14. If((n < 4) ∨ (9 < n), Segment[(0, 0), (dist, 0)], Segment[(0, 0), ((-dist), 0)]) → hiddenline1
15. Segment((0, 0), (cos(angleR), sin(angleR))) → hiddenline2
16. If(n ≤ 3, Angle[hiddenline1, hiddenline2], If[4 ≤ n ≤ 6, Angle[hiddenline2, hiddenline1], If[7 ≤ n ≤ 9, Angle[hiddenline1, hiddenline2], Angle[hiddenline2, hiddenline1]]]) → β
17. FractionText(Element[refractions, n]) → refractext
18. LaTeX((("\text{Determine the quadrant in which }\textcolor{#000099}{\theta=" * fractext) * "\pi}\text{ lies.}")) → text1
19. LaTeX((("\text{Determine the reference angle for }\textcolor{#000099}{\theta=" * fractext) * "\pi}.")) → text4
20. LaTeX((("\text{Determine which trigonometric function relates the distance travelled " * direction) * " }\cr\text{to the total distance travelled.}")) → text6
21. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
22. If(1 ≤ n ≤ 3, LaTeX[(((("\text{Since }\textcolor{#000099}{0 \le " * fractext) * "\pi \le \frac{\pi}{2}}\text{ (or, after multiplying}\cr\text{radians by }\textcolor{#000099}{\frac{180^{\circ}}{\pi \text{ radians}}}\text{ to get degrees, }\cr\textcolor{#000099}{0^{\circ}\le ") * angleD) * "^{\circ} \le 90^{\circ}}\text{), the angle is in the first quadrant.}\cr\text{The plane is flying in a northeasterly direction.}")], If[4 ≤ n ≤ 6, LaTeX[(((("\text{Since }\textcolor{#000099}{\frac{\pi}{2} < " * fractext) * "\pi \le \pi }\text{ (or, after multiplying}\cr\text{radians by }\textcolor{#000099}{\frac{180^{\circ}}{\pi \text{ radians}}}\text{ to get degrees, }\cr\textcolor{#000099}{ 90^{\circ} < ") * angleD) * "^{\circ} \le 180^{\circ}}\text{), the angle is in the second quadrant.}\cr\text{The plane is flying in a northwesterly direction.}")], If[7 ≤ n ≤ 9, LaTeX[(((("\text{Since }\textcolor{#000099}{ \pi < " * fractext) * "\pi \le \frac{3\pi}{2} }\text{ (or, after multiplying}\cr\text{radians by }\textcolor{#000099}{\frac{180^{\circ}}{\pi \text{ radians}}}\text{ to get degrees, }\cr\textcolor{#000099}{ 180^{\circ} < ") * angleD) * "^{\circ} \le 270^{\circ}}\text{), the angle is in the third quadrant.}\cr\text{The plane is flying in a southwesterly direction.}")], LaTeX[(((("\text{Since }\textcolor{#000099}{ \frac{3\pi}{2} < " * fractext) * "\pi \le 2\pi }\text{ (or, after multiplying}\cr\text{radians by }\textcolor{#000099}{\frac{180^{\circ}}{\pi \text{ radians}}}\text{ to get degrees, }\cr\textcolor{#000099}{ 270^{\circ} < ") * angleD) * "^{\circ} \le 360^{\circ}}\text{), the angle is in the fourth quadrant.}\cr\text{The plane is flying in a southeasterly direction.}")]]]) → text2
23. LaTeX((((((("\text{The direction is }\textcolor{#000099}{\theta=" * fractext) * "\pi}\text{ from east.}\cr\text{The distance is }\textcolor{#006633}{") * dist) * "\text{ miles ") * direction) * "}.}")) → Problem
24. LaTeX((((("\text{Determine the total miles the airplane has traveled}\text{ after traveling}\cr\textcolor{#006633}{\text{" * dist) * " miles ") * direction) * "}.}")) → text3
25. LaTeX((("\text{Use }\textcolor{#CC0000}{\cos\left(" * refractext) * "\pi\right)}\text{ to calculate the total distance.}")) → text8
26. LaTeX((("\text{The distance " * direction) * " is }\textcolor{#006633}{adjacent}\text{ to the reference angle. }\cr\text{The total distance traveled is the }\textcolor{#000099}{hypotenuse}\text{ of the reference triangle. }\cr\text{We know that }\textcolor{#CC0000}{ \cos(\theta)=\frac{\textcolor{#006633}\text{adjacent}}{\textcolor{#000099}{\text{hypotenuse}}}}.")) → text7
27. LaTeX((((((((((("\textcolor{#CC0000}{ \cos(\theta)}\text{ =}\frac{\textcolor{#006633}\text{adjacent}}{\textcolor{#000099}{\text{hypotenuse}}} \cr\textcolor{#CC0000}{ \cos(\theta)\textcolor{#000099}{\text{(hypotenuse)}}\text{ =}\frac{\textcolor{#006633}{\text{adjacent}}}{\textcolor{#000099}{\text{hypotenuse}}}\textcolor{#000099}{\text{(hypotenuse)}}} \cr\textcolor{#CC0000}{\cos(\theta)\textcolor{#000099}{\text{(hypotenuse)}}=\textcolor{#006633}{\text{adjacent}}} \cr\textcolor{#CC0000}{\cos(\theta)\textcolor{#000099}{\text{(hypotenuse)}}\frac{1}{\cos(\theta)}=\textcolor{#006633}{\text{adjacent}}\frac{1}{\cos(\theta)}} \cr\textcolor{#000099}{\text{hypotenuse}=\frac{\textcolor{#006633}{\text{adjacent}}}{\textcolor{#CC0000}{\cos(\theta)}}} \cr\textcolor{#000099}{\text{hypotenuse}=\frac{\textcolor{#006633}{" * dist) * "\text{ miles}}}{\textcolor{#CC0000}{\cos\left(") * refractext) * "\pi\right)}}} \cr\textcolor{#000099}{\text{hypotenuse} = \frac{\textcolor{#006633}{") * dist) * " \text{ miles}}}{\textcolor{#CC0000}{") * Element[cosval, n]) * "}}} \cr\textcolor{#000099}{\text{hypotenuse}=") * travel) * "\text{ miles}}")) → text9
28. LaTeX((("\text{The airplane traveled }\textcolor{#000099}{" * travel) * " \,miles}\text{.}")) → text10
29. If(1 ≤ n ≤ 3, LaTeX[(("\text{Since }\textcolor{#000099}{\theta= " * fractext) * "\pi }\text{ is in the first quadrant,}\cr\text{the angle is its own reference angle.}")], If[4 ≤ n ≤ 6, LaTeX[(((("\text{Since }\textcolor{#000099}{\theta= " * fractext) * "\pi }\text{ is in the second quadrant, the reference angle is }\textcolor{#CC0000}{") * refractext) * "\pi}\cr\text{measured up from the negative }\textit{x}\text{-axis as shown.}")], If[7 ≤ n ≤ 9, LaTeX[(((("\text{Since }\textcolor{#000099}{ \theta= " * fractext) * "\pi }\text{ is in the third quadrant, the reference angle is }\textcolor{#CC0000}{") * refractext) * "\pi}\cr\text{measured down from the negative }\textit{x}\text{-axis as shown.}")], LaTeX[(((("\text{Since }\textcolor{#000099}{ \theta= " * fractext) * "\pi }\text{ is in the fourth quadrant, the reference angle is }\textcolor{#CC0000}{") * refractext) * "\pi}\cr\text{measured down from the positive }\textit{x}\text{-axis as shown.}")]]]) → text5