Inverse Trigonometric Functions - AP Precalculus

1. What is the inverse of $\text{sin}(\theta) = x$?

   $\theta = \text{arcsin}(x)$. Solving for $\theta$ when sine equals $x$.

2. What is the range of the inverse sine function, $\text{arcsin}(x)$?

   The range is $[-\frac{\text{π}}{2}, \frac{\text{π}}{2}]$. Restricted to quadrants I and IV where sine is invertible.

3. Find $\text{arccot}(\text{√3})$.

   $\frac{\text{π}}{6}$. Since $\cot(\frac{\pi}{6}) = \sqrt{3}$.

4. What is $\text{arctan}(\text{√3})$?

   $\frac{\text{π}}{3}$. Since $\tan(\frac{\pi}{3}) = \sqrt{3}$.

5. Find the value of $\text{arctan}(-1)$.

   $-\frac{\text{π}}{4}$. Since $\tan(-\frac{\pi}{4}) = -1$.

6. Find $\text{arcsin}(-\frac{\text{√2}}{2})$.

   $-\frac{\text{π}}{4}$. Since $\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$.

7. What is the inverse of $\text{tan}(\theta) = x$?

   $\theta = \text{arctan}(x)$. Solving for $\theta$ when tangent equals $x$.

8. What is the range of the inverse cosecant function, $\text{arccsc}(x)$?

   $[-\frac{\text{π}}{2}, 0) \text{ or } (0, \frac{\text{π}}{2}]$. Excludes 0 where cosecant is undefined.

9. Evaluate $\text{arccos}(1)$.

   $0$. Since $\cos(0) = 1$.

10. What is $\text{arccos}(\frac{1}{2})$?

    $\frac{\text{π}}{3}$. Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

11. Find $\text{arccsc}(\text{√2})$.

    $\frac{\text{π}}{4}$. Since $\csc(\frac{\pi}{4}) = \sqrt{2}$.

12. Find $\text{arccos}(-\frac{\text{√3}}{2})$.

    $\frac{5\text{π}}{6}$. Since $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$ in quadrant II.

13. Evaluate $\text{arccot}(-1)$.

    $\frac{3\text{π}}{4}$. Since $\cot(\frac{3\pi}{4}) = -1$.

14. What is the domain of the inverse cosine function, $\text{arccos}(x)$?

    The domain is $[-1, 1]$. Cosine input values must be between -1 and 1.

15. Evaluate $\text{arcsec}(2)$.

    $\frac{\text{π}}{3}$. Since $\sec(\frac{\pi}{3}) = 2$.

16. What is $\text{arcsec}(1)$?

    $0$. Since $\sec(0) = 1$.

17. Evaluate $\text{arcsec}(-2)$.

    $\frac{2\text{π}}{3}$. Since $\sec(\frac{2\pi}{3}) = -2$ in quadrant II.

18. What is $\text{arccot}(0)$?

    $\frac{\text{π}}{2}$. Since $\cot(\frac{\pi}{2}) = 0$.

19. What is the value of $\text{arccos}(0)$?

    $\frac{\text{π}}{2}$. Since $\cos(\frac{\pi}{2}) = 0$.

20. What is the domain of the inverse secant function, $\text{arcsec}(x)$?

    $(-\text{∞}, -1] \text{ or } [1, \text{∞})$. Secant undefined where cosine equals zero.

21. Find the value of $\text{arcsin}(0)$.

    $0$. Since $\sin(0) = 0$.

22. Evaluate $\text{arccsc}(-2)$.

    $-\frac{\text{π}}{6}$. Since $\csc(-\frac{\pi}{6}) = -2$.

23. What is the range of the inverse tangent function, $\text{arctan}(x)$?

    The range is $(-\frac{\text{π}}{2}, \frac{\text{π}}{2})$. Asymptotic limits as x approaches $±∞$.

24. Evaluate $\text{arctan}(1)$.

    $\frac{\text{π}}{4}$. Since $\tan(\frac{\pi}{4}) = 1$.

25. Identify the principal value of $\text{arccos}(-1)$.

    $\text{π}$. Since $\cos(\pi) = -1$.

26. Find $\text{arcsin}(\frac{\text{√3}}{2})$.

    $\frac{\text{π}}{3}$. Since $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.

27. Evaluate $\text{arctan}(0)$.

    $0$. Since $\tan(0) = 0$.

28. What is the range of the inverse cotangent function, $\text{arccot}(x)$?

    $(0, \text{π})$. Full range from 0 to $\pi$ for all real inputs.

29. What is the domain of the inverse cotangent function, $\text{arccot}(x)$?

    $(-\text{∞}, \text{∞})$. Cotangent is defined for all real numbers.

30. Identify the principal value of $\text{arcsin}(-1)$.

    $-\frac{\text{π}}{2}$. Since $\sin(-\frac{\pi}{2}) = -1$.

31. Find $\text{arccos}(-\frac{\sqrt{3}}{2})$.

    $\frac{5\pi}{6}$. Since $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$ in quadrant II.

32. What is the principal value of $\text{arccsc}(-1)$?

    $-\frac{\text{π}}{2}$. Since $\csc(-\frac{\pi}{2}) = -1$.

33. What is the principal value of $\text{arcsec}(-1)$?

    $\text{π}$. Since $\sec(\pi) = -1$.

34. What is the principal value of $\text{arccsc}(1)$?

    $\frac{\text{π}}{2}$. Since $\csc(\frac{\pi}{2}) = 1$.

35. What is the inverse of $\text{cos}(\theta) = x$?

    $\theta = \text{arccos}(x)$. Solving for $\theta$ when cosine equals $x$.

36. What is $\text{arcsin}(\frac{1}{2})$?

    $\frac{\text{π}}{6}$. Since $\sin(\frac{\pi}{6}) = \frac{1}{2}$.

37. Evaluate $\text{arccos}(-\frac{1}{2})$.

    $\frac{2\text{π}}{3}$. Since $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$ in quadrant II.

38. Find $\text{arcsin}(-\frac{1}{2})$.

    $-\frac{\text{π}}{6}$. Since $\sin(-\frac{\pi}{6}) = -\frac{1}{2}$.

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