Trigonometric Equations and Inequalities - AP Precalculus

1. Solve $\text{tan}(\theta) = \frac{1}{\sqrt{3}}$ in $[0, 2\pi)$.

   $\theta = \frac{\pi}{6}, \frac{7\pi}{6}$. Since $\tan(\theta) = \frac{1}{\sqrt{3}}$, we have $\theta = \frac{\pi}{6}$ and $\frac{7\pi}{6}$.

2. Find the solutions for $\cot(\theta) = 0$ in $[0, 2\pi)$.

   $\theta = \frac{\pi}{2}, \frac{3\pi}{2}$. Cotangent equals zero when cosine is zero.

3. State the double angle formula for $\text{cos}(2\theta)$.

   $\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$. This is the standard double angle formula for cosine.

4. Solve $\text{sin}(2\theta) = 0$ for $\theta$ in $[0, 2\text{π})$.

   $\theta = 0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2}$. When $\sin(2\theta) = 0$, then $2\theta = n\text{π}$, so $\theta = \frac{n\text{π}}{2}$.

5. What is the period of $\text{cos}(2\theta)$?

   $\text{π}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

6. What is the general solution for $\text{csc}(\theta) = a$?

   $\theta = \text{csc}^{-1}(a) + 2n\text{π} \text{ or } -\text{csc}^{-1}(a) + 2n\text{π}$. Cosecant is symmetric, giving two solutions per period.

7. Find the solutions for $\text{cot}(\theta) = 0$ in $[0, 2\text{π})$.

   $\theta = \frac{\text{π}}{2}, \frac{3\text{π}}{2}$. Cotangent equals zero when cosine is zero.

8. Solve $\text{cos}^2(\theta) = \frac{1}{2}$ for $\theta$ in $[0, 2\text{π})$.

   $\theta = \frac{\text{π}}{4}, \frac{3\text{π}}{4}, \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Taking square root gives $\cos(\theta) = \pm\frac{\sqrt{2}}{2}$, yielding four solutions.

9. What is the general solution for $\text{sec}(\theta) = a$?

   $\theta = \text{sec}^{-1}(a) + 2n\text{π} \text{ or } -\text{sec}^{-1}(a) + 2n\text{π}$. Secant is symmetric about the y-axis, giving two solutions per period.

10. Find the solutions for $\text{sin}(\theta) = 1$ in $[0, 2\text{π})$.

    $\theta = \frac{\text{π}}{2}$. Sine equals 1 only at $\frac{\text{π}}{2}$ in the interval $[0, 2\text{π})$.

11. What is the period of $\tan(3\theta)$?

    $\frac{\text{π}}{3}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

12. What is the period of $\text{csc}(\theta)$?

    $2\text{π}$. Cosecant has the same period as sine, which is $2\text{π}$.

13. Find $\theta$ if $\text{sec}(\theta) = 2$ in $[0, 2\text{π})$.

    $\theta = \frac{\text{π}}{3}, \frac{5\text{π}}{3}$. Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we need $\cos(\theta) = \frac{1}{2}$.

14. Solve $\text{cos}(\theta) = \frac{1}{2}$ in $[0, 2\text{π})$.

    $\theta = \frac{\text{π}}{3}, \frac{5\text{π}}{3}$. Cosine equals $\frac{1}{2}$ at angles $\frac{\text{π}}{3}$ and $\frac{5\text{π}}{3}$ in $[0, 2\text{π})$.

15. Solve $\tan(\theta) = \frac{1}{\sqrt{3}}$ in $[0, 2\pi)$.

    $\theta = \frac{\pi}{6}, \frac{7\pi}{6}$. Since $\tan(\theta) = \frac{1}{\sqrt{3}}$, we have $\theta = \frac{\pi}{6}$ and $\frac{7\pi}{6}$.

16. Find the solutions for $\text{sin}(\theta) = -\frac{\text{√2}}{2}$ in $[0, 2\text{π})$.

    $\theta = \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Sine equals $-\frac{\sqrt{2}}{2}$ in the third and fourth quadrants.

17. What is the period of $\text{cot}(\theta)$?

    $\text{π}$. Cotangent function repeats its values every $\text{π}$ radians.

18. What is the period of $\text{csc}(2\theta)$?

    $\text{π}$. Period of $\csc(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

19. What is the general solution for $\text{sin}(\theta) = a$?

    $\theta = \text{sin}^{-1}(a) + 2n\text{π} \text{ or } \text{π} - \text{sin}^{-1}(a) + 2n\text{π}$. Sine has two solutions per period due to its symmetry properties.

20. What is the period of $\text{sin}(4\theta)$?

    $\frac{\text{π}}{2}$. Period of $\sin(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

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

    $\text{π}$. Tangent function repeats its values every $\text{π}$ radians.

22. State the identity for $\text{csc}(\theta)$ in terms of $\text{sin}(\theta)$.

    $\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}$. Cosecant is the reciprocal of sine.

23. Solve $\sin(\theta) = \frac{\sqrt{3}}{2}$ in $[0, 2\pi)$.

    $\theta = \frac{\pi}{3}, \frac{2\pi}{3}$. Sine equals $\frac{\sqrt{3}}{2}$ at angles $\frac{\pi}{3}$ and $\frac{2\pi}{3}$.

24. State the Pythagorean identity for $\text{tan}^2(\theta)$.

    $1 + \tan^2(\theta) = \text{sec}^2(\theta)$. This is the fundamental Pythagorean identity involving tangent and secant.

25. What is the period of $\text{tan}(4\theta)$?

    $\frac{\text{π}}{4}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{4}$ for $b=4$.

26. Find the solutions for $\text{cos}(\theta) = -\frac{1}{2}$ in $[0, 2\text{π})$.

    $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Cosine equals $-\frac{1}{2}$ at angles $\frac{2\text{π}}{3}$ and $\frac{4\text{π}}{3}$.

27. What is the period of $\text{cot}(3\theta)$?

    $\frac{\text{π}}{3}$. Period of $\cot(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

28. What is the general solution for $\text{cos}(\theta) = a$?

    $\theta = \text{cos}^{-1}(a) + 2n\text{π} \text{ or } -\text{cos}^{-1}(a) + 2n\text{π}$. Cosine is symmetric about the y-axis, giving two solutions per period.

29. Find the solutions for $\text{tan}(\theta) = 1$ in $[0, 2\text{π})$.

    $\theta = \frac{\text{π}}{4}, \frac{5\text{π}}{4}$. Tangent equals 1 at $\frac{\text{π}}{4}$ and repeats every $\text{π}$.

30. Find the solutions for $\text{csc}(\theta) = -2$ in $[0, 2\text{π})$.

    $\theta = \frac{7\text{π}}{6}, \frac{11\text{π}}{6}$. Since $\csc(\theta) = -2$, we need $\sin(\theta) = -\frac{1}{2}$.

31. What is the period of $\text{cos}(4\theta)$?

    $\frac{\text{π}}{2}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

32. What is the amplitude of $\text{sin}(3\theta)$?

    1. Amplitude is the coefficient of the sine function, which is 1.

33. What is the general solution for $\text{sin}(\theta) = 0$?

    $\theta = n\text{π}$, $n \text{ is an integer}$. Sine equals zero at multiples of $\text{π}$.

34. Find $\theta$ if $\sec(\theta) = 2$ in $[0, 2\pi)$.

    $\theta = \frac{\pi}{3}, \frac{5\pi}{3}$. Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we need $\cos(\theta) = \frac{1}{2}$.

35. What is the period of $\text{sec}(\theta)$?

    $2\text{π}$. Secant has the same period as cosine, which is $2\text{π}$.

36. What is the period of $\text{sin}(\theta)$?

    $2\text{π}$. Sine function completes one cycle every $2\text{π}$ radians.

37. State the identity for $\text{sec}(\theta)$ in terms of $\text{cos}(\theta)$.

    $\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}$. Secant is the reciprocal of cosine.

38. Find the solutions for $\text{cos}(\theta) = 0$ in $[0, 2\text{π})$.

    $\theta = \frac{\text{π}}{2}, \frac{3\text{π}}{2}$. Cosine equals zero when the angle is an odd multiple of $\frac{\text{π}}{2}$.

39. Find the solutions for $\text{sec}(\theta) = -2$ in $[0, 2\text{π})$.

    $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Since $\sec(\theta) = -2$, we need $\cos(\theta) = -\frac{1}{2}$.

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