AP Precalculus Flashcards: Inverse Trigonometric Functions

What this deck covers

This deck focuses on Inverse Trigonometric Functions, 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 inverse of $\text{sin}(\theta) = x$?

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

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

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

Flashcard 3: Find $\text{arccot}(\text{√3})$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 11: Find $\text{arccsc}(\text{√2})$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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