AP Precalculus Flashcards: The Tangent Function

What this deck covers

This deck focuses on The Tangent Function, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the tangent of 45 degrees?

Answer:

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

Flashcard 2: What is the tangent of 45 degrees?

Answer:

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

Flashcard 3: Find the tangent of $\frac{7\pi}{4}$ radians.

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

Flashcard 4: What is $\tan(\pi - x)$ in terms of $\tan(x)$?

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

Flashcard 5: What is the fundamental period of the tangent function?

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

Flashcard 6: Find $\tan(x - \pi)$ in terms of $\tan(x)$.

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

Flashcard 7: What is the tangent of 30 degrees?

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

Flashcard 8: What is the integral of $\tan(x)$?

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

Flashcard 9: What is the range of the tangent function?

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

Flashcard 10: Find $\tan(2\pi + x)$ in terms of $\tan(x)$.

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

Flashcard 11: What is the tangent of 30 degrees?

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

Flashcard 12: Find the tangent of $\frac{\pi}{4}$ radians.

Answer:

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

Flashcard 13: Find $\tan(\pi + x)$ in terms of $\tan(x)$.

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

Flashcard 14: Find $\tan(-x)$ in terms of $\tan(x)$.

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

Flashcard 15: What is the definition of the tangent function in a right triangle?

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

Flashcard 16: Find the tangent of $\frac{\pi}{3}$ radians.

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

Flashcard 17: Find the tangent of $\frac{5\pi}{6}$ radians.

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

Flashcard 18: What is the definition of the tangent function in a right triangle?

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

Flashcard 19: Find the value of $\tan(\pi)$.

Answer:

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

Flashcard 20: Find the tangent of $\frac{4\pi}{3}$ radians.

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

Flashcard 21: Identify the symmetry type of the tangent function.

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

Flashcard 22: What is the period of the tangent function?

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

Flashcard 23: Find the tangent of $\frac{7\pi}{6}$ radians.

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

Flashcard 24: Find $\tan(x - \pi)$ in terms of $\tan(x)$.

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

Flashcard 25: Find the tangent of $\frac{\pi}{6}$ radians.

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

Flashcard 26: What is the integral of $\tan(x)$?

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

Flashcard 27: What is the tangent of 90 degrees?

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

Flashcard 28: Find the tangent of $\frac{\pi}{3}$ radians.

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

Flashcard 29: What is the tangent of 180 degrees?

Answer:

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

Flashcard 30: State the formula for the tangent of an angle $\theta$ in a right triangle.

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

Flashcard 31: What is the tangent of 0 degrees?

Answer:

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

Flashcard 32: What is the tangent of $2\pi$ radians?

Answer:

1. $2\pi$ radians equals $360°$, same as $0°$.

Flashcard 33: Find $\tan(\pi + x)$ in terms of $\tan(x)$.

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

Flashcard 34: Find the tangent of $\frac{\pi}{4}$ radians.

Answer:

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

Flashcard 35: Find $\tan(-x)$ in terms of $\tan(x)$.

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

Flashcard 36: Find the tangent of $\frac{5\pi}{3}$ radians.

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

Flashcard 37: What is $\tan(x + \pi)$ in terms of $\tan(x)$?

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

Flashcard 38: What is the fundamental period of the tangent function?

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

Flashcard 39: Identify the symmetry type of the tangent function.

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

Flashcard 40: What is the tangent of 180 degrees?

Answer:

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

Flashcard 41: What is the tangent of 0 degrees?

Answer:

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

Flashcard 42: What is $\tan(\pi - x)$ in terms of $\tan(x)$?

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

Flashcard 43: Determine the value of $\tan(\frac{\pi}{2} - x)$.

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

Flashcard 44: Find $\tan(2\pi + x)$ in terms of $\tan(x)$.

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

Flashcard 45: What is the tangent of 60 degrees?

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

Flashcard 46: Find the tangent of $\frac{4\pi}{3}$ radians.

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

Flashcard 47: Identify the vertical asymptotes of the tangent function.

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

Flashcard 48: Determine the value of $\tan(\frac{\pi}{2} - x)$.

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

Flashcard 49: Find the tangent of $270$ degrees.

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

Flashcard 50: Find the value of $\tan(\pi)$.

Answer:

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

Flashcard 51: What is $\tan(x + \pi)$ in terms of $\tan(x)$?

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

Flashcard 52: What is $\tan(x)$ in terms of sine and cosine?

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

Flashcard 53: Find the tangent of $\frac{11\pi}{6}$ radians.

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

Flashcard 54: What is the derivative of $\tan(x)$?

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

Flashcard 55: What is the range of the tangent function?

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

Flashcard 56: Find the tangent of $\frac{5\pi}{3}$ radians.

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

Flashcard 57: State the formula for the tangent of an angle $\theta$ in a right triangle.

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

Flashcard 58: What is the derivative of $\tan(x)$?

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

Flashcard 59: Find the tangent of $\frac{7\pi}{4}$ radians.

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

Flashcard 60: What is $\tan(x)$ in terms of sine and cosine?

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

Flashcard 61: Identify the vertical asymptotes of the tangent function.

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

Flashcard 62: Find the tangent of $\frac{7\pi}{6}$ radians.

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

Flashcard 63: Find the tangent of $\frac{3\pi}{4}$ radians.

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

Flashcard 64: What is the period of the tangent function?

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

Flashcard 65: Find the tangent of $\frac{\pi}{6}$ radians.

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

Flashcard 66: Find the tangent of $\frac{5\pi}{6}$ radians.

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

Flashcard 67: Find the tangent of $\frac{5\pi}{4}$ radians.

Answer:

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

Flashcard 68: What is the tangent of 90 degrees?

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

Flashcard 69: Find the tangent of $\frac{11\pi}{6}$ radians.

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

Flashcard 70: Find the tangent of $\frac{3\pi}{4}$ radians.

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

Flashcard 71: Find the tangent of $270$ degrees.

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

Flashcard 72: What is the tangent of $2\pi$ radians?

Answer:

1. $2\pi$ radians equals $360°$, same as $0°$.

Flashcard 73: Find the tangent of $\frac{5\pi}{4}$ radians.

Answer:

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

Flashcard 74: What is the tangent of 60 degrees?

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