The Tangent Function - AP Precalculus

1. What is the tangent of 45 degrees?

   1. At $45°$, opposite and adjacent sides are equal in length.

2. Find the tangent of $\frac{7\pi}{4}$ radians.

   -1. Reference angle is $\frac{\pi}{4}$ in quadrant IV where tangent is negative.

3. Find $\tan(x - \pi)$ in terms of $\tan(x)$.

   $\tan(x)$. Subtracting $\pi$ is equivalent to adding $\pi$ due to periodicity.

4. Find $\tan(2\pi + x)$ in terms of $\tan(x)$.

   $\tan(x)$. Period is $\pi$, so adding $2\pi$ doesn't change the value.

5. What is the tangent of 30 degrees?

   $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. Standard value from the 30-60-90 triangle.

6. Find $\tan(\pi + x)$ in terms of $\tan(x)$.

   $\tan(x)$. Tangent has period $\pi$, so adding $\pi$ doesn't change the value.

7. Find the tangent of $\frac{\pi}{3}$ radians.

   $\sqrt{3}$. In a 30-60-90 triangle, $\tan(60°) = \sqrt{3}$.

8. Find the tangent of $\frac{5\pi}{6}$ radians.

   $-\frac{1}{\sqrt{3}}$ or $-\frac{\sqrt{3}}{3}$. At $150°$, we're in quadrant II where tangent is negative.

9. What is the definition of the tangent function in a right triangle?

   Tangent is the ratio of the opposite side to the adjacent side. Fundamental definition for the tangent ratio in right triangles.

10. Find the value of $\tan(\pi)$.

    0. $\pi$ radians equals $180°$ where tangent is zero.

11. What is the period of the tangent function?

    $\pi$. Tangent repeats its values every $\pi$ radians.

12. What is the integral of $\tan(x)$?

    $-\ln|\cos(x)| + C$. Standard antiderivative formula for the tangent function.

13. What is the tangent of 90 degrees?

    Undefined. At $90°$, adjacent side is 0, making division undefined.

14. What is the tangent of 180 degrees?

    0. At $180°$, we're on the negative x-axis where tangent is 0.

15. What is the tangent of $2\pi$ radians?

    0. $2\pi$ radians equals $360°$, same as $0°$.

16. Find the tangent of $\frac{\pi}{4}$ radians.

    1. At $45°$, opposite and adjacent sides are equal.

17. Find $\tan(-x)$ in terms of $\tan(x)$.

    $-\tan(x)$. Tangent is an odd function, so $\tan(-x) = -\tan(x)$.

18. What is the fundamental period of the tangent function?

    $\pi$. The smallest positive period for the tangent function.

19. Identify the symmetry type of the tangent function.

    Odd function. Since $\tan(-x) = -\tan(x)$, tangent is an odd function.

20. What is the tangent of 0 degrees?

    0. At $0°$, opposite side is 0 while adjacent side is positive.

21. What is $\tan(\pi - x)$ in terms of $\tan(x)$?

    $-\tan(x)$. Using the identity $\tan(\pi - x) = -\tan(x)$.

22. Determine the value of $\tan(\frac{\pi}{2} - x)$.

    $\cot(x)$. Using the cofunction identity for complementary angles.

23. What is the tangent of 60 degrees?

    $\sqrt{3}$. Standard value from the 30-60-90 triangle.

24. Find the tangent of $\frac{4\pi}{3}$ radians.

    $\sqrt{3}$. Reference angle is $\frac{\pi}{3}$ in quadrant III.

25. Identify the vertical asymptotes of the tangent function.

    $x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$. Occurs where cosine equals zero, causing division by zero.

26. What is $\tan(x + \pi)$ in terms of $\tan(x)$?

    $\tan(x)$. Since tangent has period $\pi$, adding $\pi$ doesn't change the value.

27. What is $\tan(x)$ in terms of sine and cosine?

    $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Fundamental relationship between tangent, sine, and cosine.

28. What is the derivative of $\tan(x)$?

    $\sec^2(x)$. Standard derivative formula for the tangent function.

29. What is the range of the tangent function?

    All real numbers. Tangent can take any real value from $-\infty$ to $+\infty$.

30. Find the tangent of $\frac{5\pi}{3}$ radians.

    $-\sqrt{3}$. Reference angle is $\frac{\pi}{3}$ in quadrant IV where tangent is negative.

31. State the formula for the tangent of an angle $\theta$ in a right triangle.

    $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Basic trigonometric ratio formula using sides of a right triangle.

32. Find the tangent of $\frac{7\pi}{6}$ radians.

    $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. Reference angle is $\frac{\pi}{6}$ in quadrant III.

33. Find the tangent of $\frac{\pi}{6}$ radians.

    $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$. In a 30-60-90 triangle, $\tan(30°) = \frac{1}{\sqrt{3}}$.

34. Find the tangent of $\frac{11\pi}{6}$ radians.

    $-\frac{1}{\sqrt{3}}$ or $-\frac{\sqrt{3}}{3}$. Reference angle is $\frac{\pi}{6}$ in quadrant IV where tangent is negative.

35. Find the tangent of $\frac{3\pi}{4}$ radians.

    -1. At $135°$, we're in quadrant II where tangent is negative.

36. Find the tangent of $270$ degrees.

    Undefined. At $270°$, cosine is 0, making tangent undefined.

37. Find the tangent of $\frac{5\pi}{4}$ radians.

    1. Reference angle is $\frac{\pi}{4}$ in quadrant III.

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