Solving Trigonometric Equations in Context - Geometry

1. What is $\arctan(\tan(\theta))$ equal to in general?

   The angle in $(-\frac{\pi}{2},\frac{\pi}{2})$ coterminal with $\theta$. Returns the equivalent angle in the principal range.

2. What is the inverse-trig step to solve $\sin(\theta)=a$ when $a\in[-1,1]$?

   Start with $\theta=\arcsin(a)$ as the principal angle. Find the reference angle first, then all solutions.

3. Identify the correct degree-to-radian conversion formula for an angle $\theta$.

   $\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.

4. What is the range of $\arccos(x)$?

   $[0,\pi]$. Output angles from 0 to $\pi$ inclusive.

5. Solve for $x$ in $\cos(x)=\frac{1}{2}$ on $[0,\pi]$.

   $x=\frac{\pi}{3}$. Only one solution of $\cos(x) = \frac{1}{2}$ lies in $[0,\pi]$.

6. Solve for $x$ in $\tan(x)=-1$ on $[0,2\pi)$.

   $x=\frac{3\pi}{4},\frac{7\pi}{4}$. Negative tangent in Quadrants II and IV.

7. What is the general solution of $\tan(\theta)=\tan(\alpha)$ in radians?

   $\theta=\alpha+\pi k$. Uses tangent's $\pi$ periodicity.

8. Find all solutions in $[0,2\pi)$ for $\cos(\theta)=-\frac{1}{2}$.

   $\theta=\frac{2\pi}{3},\frac{4\pi}{3}$. Negative cosine in Quadrants II and III.

9. Identify the valid model output if an angle of elevation must satisfy $0\le\theta\le\frac{\pi}{2}$.

   Keep only solutions with $0\le\theta\le\frac{\pi}{2}$. Elevation angles must be between 0 and 90 degrees.

10. Solve for $x$ in $\cos(x)=0.2$ on $[0,2\pi)$ using inverse trig notation.

    $x=\arccos(0.2)$ or $x=2\pi-\arccos(0.2)$. General form using inverse trig for principal values.

11. What is $\arccos(\cos(\theta))$ equal to in general?

    The angle in $[0,\pi]$ coterminal with $\theta$. Returns the equivalent angle in the principal range.

12. What identity connects $\sin(\arcsin(x))$ for $x\in[-1,1]$?

    $\sin(\arcsin(x))=x$. Inverse cancels the function when input is in domain.

13. What is the principal range of $\arccos(x)$ in radians?

    $[0,\pi]$. Principal range where $\arccos$ returns unique values.

14. What identity connects $\tan(\arctan(x))$ for all real $x$?

    $\tan(\arctan(x))=x$. Inverse cancels the function for all real inputs.

15. Solve $\tan(2x)=1$ for $x$ in $[0,\pi)$.

    $x=\frac{\pi}{8},\frac{5\pi}{8}$. Set $2x = \frac{\pi}{4}, \frac{5\pi}{4}$ and solve for $x$.

16. Find all solutions in $[0,2\pi)$ for $\sin(\theta)=-\frac{1}{2}$.

    $\theta=\frac{7\pi}{6},\frac{11\pi}{6}$. Negative sine in Quadrants III and IV.

17. Solve for $x$ in $2\sin(x)-1=0$ on $[0,2\pi)$.

    $x=\frac{\pi}{6},\frac{5\pi}{6}$. Solve $\sin(x) = \frac{1}{2}$ by isolating the sine term.

18. What is the range of $\arctan(x)$?

    $(-\frac{\pi}{2},\frac{\pi}{2})$. Output angles between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ exclusive.

19. Find all solutions in $[0,2\pi)$ for $\tan(\theta)=1$.

    $\theta=\frac{\pi}{4},\frac{5\pi}{4}$. Tangent has period $\pi$, so add $\pi$ to first solution.

20. What is the domain of $\arcsin(x)$ and $\arccos(x)$?

    $[-1,1]$. Both functions require input between -1 and 1 inclusive.

21. Identify the correct radian-to-degree conversion formula for an angle $\theta$.

    $\theta_{deg}=\theta_{rad}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.

22. What is the midline of $y=A\cos(Bx+C)+D$?

    $y=D$. Vertical shift $D$ moves the center line up or down.

23. Find the principal solution for $\cos(\theta)=\frac{1}{2}$ in radians.

    $\theta=\frac{\pi}{3}$. Since $\arccos(\frac{1}{2}) = \frac{\pi}{3}$ in Quadrant I.

24. Solve $\cos(2x)=0$ for $x$ in $[0,\pi)$.

    $x=\frac{\pi}{4},\frac{3\pi}{4}$. Set $2x = \frac{\pi}{2}, \frac{3\pi}{2}$ and solve for $x$.

25. Solve for $t$ in $\cos(\pi t)=0$ on $[0,2]$.

    $t=\frac{1}{2},\frac{3}{2}$. Set $\pi t = \frac{\pi}{2}, \frac{3\pi}{2}$ and solve for $t$.

26. What is the principal range of $\arctan(x)$ in radians?

    $(-\frac{\pi}{2},\frac{\pi}{2})$. Principal range where $\arctan$ returns unique values.

27. Identify whether $\sin(\theta)=1.2$ has any real solution.

    No real solution because $1.2\notin[-1,1]$. Values outside $[-1,1]$ have no real inverse trig solutions.

28. Solve for $x$ in $\sin(x)=0.5$ on $[0,2\pi)$ using inverse trig notation.

    $x=\arcsin(0.5)$ or $x=\pi-\arcsin(0.5)$. General form using inverse trig for principal values.

29. What is the inverse-trig step to solve $\cos(\theta)=a$ when $a\in[-1,1]$?

    Start with $\theta=\arccos(a)$ as the principal angle. Find the reference angle first, then all solutions.

30. Find the principal solution for $\sin(\theta)=\frac{1}{2}$ in radians.

    $\theta=\frac{\pi}{6}$. Since $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ in Quadrant I.

31. What is the general solution of $\sin(\theta)=\sin(\alpha)$ in radians?

    $\theta=\alpha+2\pi k$ or $\theta=\pi-\alpha+2\pi k$. Uses sine's symmetry about $\frac{\pi}{2}$ and $2\pi$ periodicity.

32. What is the amplitude of $y=A\sin(Bx+C)+D$?

    $|A|$. Maximum distance from midline to peak or trough.

33. Solve for $x$ in $\sin(x)=\frac{1}{2}$ on $[0,\pi]$.

    $x=\frac{\pi}{6},\frac{5\pi}{6}$. Both solutions of $\sin(x) = \frac{1}{2}$ lie in $[0,\pi]$.

34. Find all solutions in $[0,2\pi)$ for $\sin(\theta)=0$.

    $\theta=0,\pi$. Sine equals zero at multiples of $\pi$.

35. What is the principal range of $\arcsin(x)$ in radians?

    $[-\frac{\pi}{2},\frac{\pi}{2}]$. Principal range where $\arcsin$ returns unique values.

36. What is the general solution of $\cos(\theta)=\cos(\alpha)$ in radians?

    $\theta=\alpha+2\pi k$ or $\theta=-\alpha+2\pi k$. Uses cosine's even symmetry and $2\pi$ periodicity.

37. Find the second solution in $[0,2\pi)$ for $\sin(\theta)=\frac{1}{2}$.

    $\theta=\frac{5\pi}{6}$. Since $\sin(\theta) = \frac{1}{2}$ in Quadrant II at $\pi - \frac{\pi}{6}$.

38. What is the period of $\sin(\theta)$ and $\cos(\theta)$?

    $2\pi$. Both functions complete one cycle every $2\pi$ radians.

39. What is the domain of $\arctan(x)$?

    $(-\infty,\infty)$. Tangent accepts all real number inputs.

40. What is $\arcsin(\sin(\theta))$ equal to in general?

    The angle in $[-\frac{\pi}{2},\frac{\pi}{2}]$ coterminal with $\theta$. Returns the equivalent angle in the principal range.

41. Identify the valid contextual time if a model gives $t=-2$ seconds and $t=3$ seconds.

    $t=3$ seconds. Physical time cannot be negative in most contexts.

42. What is the period of $\tan(\theta)$?

    $\pi$. Tangent completes one cycle every $\pi$ radians.

43. What identity connects $\cos(\arccos(x))$ for $x\in[-1,1]$?

    $\cos(\arccos(x))=x$. Inverse cancels the function when input is in domain.

44. What condition must hold for $\arcsin(u)$ or $\arccos(u)$ to be real?

    $-1\le u\le 1$. Both inverse functions require inputs in $[-1,1]$.

45. Find the principal solution for $\tan(\theta)=1$ in radians.

    $\theta=\frac{\pi}{4}$. Since $\arctan(1) = \frac{\pi}{4}$ in Quadrant I.

46. Solve for $t$ in $\sin(\pi t)=0$ on $[0,2]$.

    $t=0,1,2$. Set $\pi t = 0, \pi, 2\pi$ and solve for $t$.

47. Solve $\sin(2x)=0$ for $x$ in $[0,\pi)$.

    $x=0,\frac{\pi}{2}$. Set $2x = 0, \pi$ and solve for $x$.

48. Find the second solution in $[0,2\pi)$ for $\cos(\theta)=\frac{1}{2}$.

    $\theta=\frac{5\pi}{3}$. Since $\cos(\theta) = \frac{1}{2}$ in Quadrant IV at $2\pi - \frac{\pi}{3}$.

49. What is the range of $\arcsin(x)$?

    $[-\frac{\pi}{2},\frac{\pi}{2}]$. Output angles from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ inclusive.

50. What is the inverse-trig step to solve $\tan(\theta)=a$ for real $a$?

    Start with $\theta=\arctan(a)$ as the principal angle. Find the reference angle first, then all solutions.

51. What is the period of $y=A\sin(Bx+C)+D$ in radians?

    $\frac{2\pi}{|B|}$. Coefficient $B$ affects how often the function repeats.

52. What is the phase shift of $y=A\sin(Bx+C)+D$ written in that form?

    $-\frac{C}{B}$. Horizontal shift determined by $-\frac{C}{B}$ formula.

53. Find all solutions in $[0,2\pi)$ for $\cos(\theta)=0$.

    $\theta=\frac{\pi}{2},\frac{3\pi}{2}$. Cosine equals zero at odd multiples of $\frac{\pi}{2}$.

54. Solve for $x$ in $\tan(x)=1$ on $[0,\pi]$.

    $x=\frac{\pi}{4}$. Only one solution of $\tan(x) = 1$ lies in $[0,\pi]$.

55. Solve for $x$ in $3\cos(x)+3=0$ on $[0,2\pi)$.

    $x=\pi$. Solve $\cos(x) = -1$ by isolating the cosine term.

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