Constructing Inverse Trigonometric Functions - Geometry

1. What monotonic behavior on an interval ensures a function is one-to-one there?

   Always increasing or always decreasing. Monotonic functions are one-to-one since they never repeat outputs.

2. What is the range of $y=\arctan(x)$?

   $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. The range of an inverse function equals the domain of the original function.

3. Find the restricted interval where $\sin(x)$ is decreasing on $\left[\frac{\pi}{2},\frac{3\pi}{2}\right]$.

   $\left[\frac{\pi}{2},\frac{3\pi}{2}\right]$. The sine function is decreasing throughout this entire interval.

4. What is the principal restricted domain used to define $y=\arcsin(x)$?

   $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. This interval makes $\sin(x)$ strictly increasing and one-to-one.

5. Evaluate $\tan\left(\arctan(x)\right)$ for $x\in\left(-\infty,\infty\right)$.

   $x$. Composing a function with its inverse yields the original input.

6. Identify whether $\sin(x)$ is one-to-one on $\left[0,2\pi\right]$.

   No, it is not one-to-one on $\left[0,2\pi\right]$. It fails the horizontal line test due to periodicity.

7. What is the value of $\arccos\left(\frac{1}{2}\right)$ in the principal range?

   $\frac{\pi}{3}$. $\cos(\frac{\pi}{3}) = \frac{1}{2}$, so $\arccos(\frac{1}{2}) = \frac{\pi}{3}$.

8. Choose the correct statement: $\arccos(x)$ outputs angles in which interval?

   $\left[0,\pi\right]$. Arccosine outputs angles in its principal range from $0$ to $\pi$.

9. Identify whether $\sin(x)$ is one-to-one on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

   Yes, it is one-to-one on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. It passes the horizontal line test on this restricted interval.

10. Identify whether $\cos(x)$ is one-to-one on $\left[0,2\pi\right]$.

    No, it is not one-to-one on $\left[0,2\pi\right]$. It fails the horizontal line test due to periodicity.

11. Identify whether $\tan(x)$ is one-to-one on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

    Yes, it is one-to-one on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. It passes the horizontal line test on this restricted interval.

12. What is the value of $\arctan(1)$ in the principal range?

    $\frac{\pi}{4}$. $\tan(\frac{\pi}{4}) = 1$, so $\arctan(1) = \frac{\pi}{4}$.

13. Evaluate $\arcsin\left(\sin\left(\frac{\pi}{3}\right)\right)$.

    $\frac{\pi}{3}$. $\frac{\pi}{3}$ is in the principal range of arcsine.

14. Which inverse trig function returns an angle in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?

    $\arcsin(x)$. The arcsine function outputs angles in its principal range.

15. Identify whether $\cos(x)$ is one-to-one on $\left[0,\pi\right]$.

    Yes, it is one-to-one on $\left[0,\pi\right]$. It passes the horizontal line test on this restricted interval.

16. What is the value of $\arctan(-\sqrt{3})$ in the principal range?

    $-\frac{\pi}{3}$. $\tan(-\frac{\pi}{3}) = -\sqrt{3}$, so $\arctan(-\sqrt{3}) = -\frac{\pi}{3}$.

17. What is the principal value of $\arctan\left(\tan\left(\frac{5\pi}{6}\right)\right)$?

    $-\frac{\pi}{6}$. $\tan(\frac{5\pi}{6}) = -\frac{1}{\sqrt{3}}$, so $\arctan(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}$.

18. Evaluate $\arccos\left(\cos\left(\frac{4\pi}{3}\right)\right)$.

    $\frac{2\pi}{3}$. $\cos(\frac{4\pi}{3}) = \cos(\frac{2\pi}{3})$, and $\frac{2\pi}{3}$ is in the principal range.

19. Evaluate $\arctan\left(\tan\left(-\frac{\pi}{4}\right)\right)$.

    $-\frac{\pi}{4}$. $-\frac{\pi}{4}$ is in the principal range of arctangent.

20. What is the range of $y=\arccos(x)$?

    $\left[0,\pi\right]$. The range of an inverse function equals the domain of the original function.

21. What is the value of $\arccos\left(-\frac{\sqrt{2}}{2}\right)$ in the principal range?

    $\frac{3\pi}{4}$. $\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$, so $\arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}$.

22. On what interval is $\cos(x)$ strictly decreasing, making it invertible there?

    $\left[0,\pi\right]$. On this interval, $\cos(x)$ decreases from $1$ to $-1$ without repeating.

23. What is the range of $\sin(x)$ on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?

    $\left[-1,1\right]$. The sine function achieves all values from $-1$ to $1$ on this restricted domain.

24. What is the domain of $y=\arcsin(x)$?

    $\left[-1,1\right]$. The domain of an inverse function equals the range of the original function.

25. What is the principal restricted domain used to define $y=\arccos(x)$?

    $\left[0,\pi\right]$. This interval makes $\cos(x)$ strictly decreasing and one-to-one.

26. What is the domain of $y=\arctan(x)$?

    $\left(-\infty,\infty\right)$. The domain of an inverse function equals the range of the original function.

27. What is the range of $\cos(x)$ on $\left[0,\pi\right]$?

    $\left[-1,1\right]$. The cosine function achieves all values from $-1$ to $1$ on this restricted domain.

28. Identify the principal value of $\arccos\left(\cos\left(-\frac{\pi}{3}\right)\right)$.

    $\frac{\pi}{3}$. $\cos(-\frac{\pi}{3}) = \cos(\frac{\pi}{3})$, and $\frac{\pi}{3}$ is in the principal range.

29. What does it mean to restrict a function to construct an inverse?

    Limit its domain to make it one-to-one. Restricting eliminates repeated outputs to ensure invertibility.

30. What relationship holds for inverse functions $f$ and $f^{-1}$?

    $f\left(f^{-1}(x)\right)=x$ on the domain of $f^{-1}$. This is the fundamental property defining inverse functions.

31. What property must a function have on a restricted domain to have an inverse function?

    It must be one-to-one on that domain. One-to-one means each output corresponds to exactly one input.

32. Which inverse trig function returns an angle in $\left[0,\pi\right]$?

    $\arccos(x)$. The arccosine function outputs angles in its principal range.

33. Find the restricted interval where $\cos(x)$ is increasing on $\left[\pi,2\pi\right]$.

    $\left[\pi,2\pi\right]$. The cosine function is increasing throughout this entire interval.

34. Choose the correct statement: $\arcsin(x)$ is defined only for which $x$-values?

    $x\in\left[-1,1\right]$. Arcsine is only defined for inputs between $-1$ and $1$.

35. Evaluate $\sin\left(\arcsin(x)\right)$ for $x\in\left[-1,1\right]$.

    $x$. Composing a function with its inverse yields the original input.

36. On what interval is $\tan(x)$ strictly increasing and commonly restricted to define an inverse?

    $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. On this interval, $\tan(x)$ increases from $-\infty$ to $\infty$ without repeating.

37. What is the range of $\tan(x)$ on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$?

    $\left(-\infty,\infty\right)$. The tangent function achieves all real values on this restricted domain.

38. What is the value of $\arcsin\left(-\frac{\sqrt{2}}{2}\right)$ in the principal range?

    $-\frac{\pi}{4}$. $\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$, so $\arcsin(-\frac{\sqrt{2}}{2}) = -\frac{\pi}{4}$.

39. Identify the principal value of $\arcsin\left(\sin\left(-\frac{3\pi}{4}\right)\right)$.

    $-\frac{\pi}{4}$. $\sin(-\frac{3\pi}{4}) = -\sin(\frac{\pi}{4})$, giving $-\frac{\pi}{4}$ in the principal range.

40. What test guarantees a function is one-to-one on an interval and thus invertible there?

    The horizontal line test on that interval. If no horizontal line intersects the graph twice, the function is one-to-one.

41. Evaluate $\arcsin\left(\sin\left(\frac{2\pi}{3}\right)\right)$.

    $\frac{\pi}{3}$. $\sin(\frac{2\pi}{3}) = \sin(\frac{\pi}{3})$, and $\frac{\pi}{3}$ is in the principal range.

42. What is the domain of $y=\arccos(x)$?

    $\left[-1,1\right]$. The domain of an inverse function equals the range of the original function.

43. On what interval is $\sin(x)$ strictly increasing, making it invertible there?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. On this interval, $\sin(x)$ increases from $-1$ to $1$ without repeating.

44. Evaluate $\cos\left(\arccos(x)\right)$ for $x\in\left[-1,1\right]$.

    $x$. Composing a function with its inverse yields the original input.

45. What is the value of $\arcsin\left(\frac{1}{2}\right)$ in the principal range?

    $\frac{\pi}{6}$. $\sin(\frac{\pi}{6}) = \frac{1}{2}$, so $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$.

46. What is the range of $y=\arcsin(x)$?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. The range of an inverse function equals the domain of the original function.

47. Evaluate $\arccos\left(\cos\left(\frac{2\pi}{3}\right)\right)$.

    $\frac{2\pi}{3}$. $\frac{2\pi}{3}$ is in the principal range of arccosine.

48. Which inverse trig function returns an angle in $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$?

    $\arctan(x)$. The arctangent function outputs angles in its principal range.

49. What is the principal restricted domain used to define $y=\arctan(x)$?

    $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. This interval makes $\tan(x)$ strictly increasing and one-to-one.

50. What geometric transformation relates the graph of $f$ and the graph of $f^{-1}$?

    Reflection across the line $y=x$. Inverse functions are reflections of each other across this diagonal line.

51. What relationship holds for inverse functions $f$ and $f^{-1}$ (other composition order)?

    $f^{-1}(f(x))=x$ on the restricted domain of $f$. This is the other fundamental property defining inverse functions.

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