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

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

Flashcard 4: 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 5: 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 6: 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 7: Find the tangent of $\frac{\pi}{3}$ radians.

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

Flashcard 8: 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 9: 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 10: Find the value of $\tan(\pi)$.

Answer:

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

Flashcard 11: What is the period of the tangent function?

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

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

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

Flashcard 13: What is the tangent of 90 degrees?

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

Flashcard 14: What is the tangent of 180 degrees?

Answer:

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

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

Answer:

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

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

Answer:

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

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

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

Flashcard 18: What is the fundamental period of the tangent function?

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

Flashcard 19: Identify the symmetry type of the tangent function.

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

Flashcard 20: What is the tangent of 0 degrees?

Answer:

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

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

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

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

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

Flashcard 23: What is the tangent of 60 degrees?

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

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

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

Flashcard 25: 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 26: 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 27: 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 28: What is the derivative of $\tan(x)$?

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

Flashcard 29: What is the range of the tangent function?

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

Flashcard 30: 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 31: 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 32: 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 33: 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 34: 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 35: Find the tangent of $\frac{3\pi}{4}$ radians.

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

Flashcard 36: Find the tangent of $270$ degrees.

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

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

Answer:

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