AP Precalculus Flashcards: Inverse Trigonometric Functions

Study Inverse Trigonometric Functions in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Inverse Trigonometric Functions

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QUESTION

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What is the inverse of $\text{sin}(\theta) = x$ ?

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ANSWER

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

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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.

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

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

Flashcard 4: Find $\text{arccot}(\text{√3})$ .

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

Flashcard 5: What is $\text{arctan}(\text{√3})$ ?

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

Flashcard 6: What is the inverse of $\text{tan}(\theta) = x$ ?

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

Flashcard 7: What is $\text{arcsec}(1)$ ?

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

Flashcard 8: Find the value of $\text{arctan}(-1)$ .

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

Flashcard 9: 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 10: Find $\text{arcsin}(-\frac{\text{√2}}{2})$ .

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

Flashcard 11: What is the inverse of $\text{tan}(\theta) = x$ ?

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

Flashcard 12: Evaluate $\text{arccot}(-1)$ .

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

Flashcard 13: 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 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 $\text{arccos}(1)$ .

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

Flashcard 16: What is $\text{arccos}(\frac{1}{2})$ ?

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

Flashcard 17: Find the value of $\text{arcsin}(0)$ .

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

Flashcard 18: Find $\text{arccsc}(\text{√2})$ .

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

Flashcard 19: What is the inverse of $\text{cos}(\theta) = x$ ?

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

Flashcard 20: 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 21: What is $\text{arccot}(0)$ ?

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

Flashcard 22: Evaluate $\text{arccos}(1)$ .

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

Flashcard 23: Evaluate $\text{arccot}(-1)$ .

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

Flashcard 24: 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 25: 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 26: Identify the principal value of $\text{arccos}(-1)$ .

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

Flashcard 27: Identify the principal value of $\text{arcsin}(-1)$ .

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

Flashcard 28: Evaluate $\text{arcsec}(2)$ .

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

Flashcard 29: What is the principal value of $\text{arccsc}(1)$ ?

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

Flashcard 30: What is $\text{arcsec}(1)$ ?

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

Flashcard 31: Evaluate $\text{arcsec}(-2)$ .

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

Flashcard 32: 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 33: What is $\text{arccot}(0)$ ?

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

Flashcard 34: What is the value of $\text{arccos}(0)$ ?

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

Flashcard 35: 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 36: Find the value of $\text{arctan}(-1)$ .

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

Flashcard 37: Find the value of $\text{arcsin}(0)$ .

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

Flashcard 38: Evaluate $\text{arccsc}(-2)$ .

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

Flashcard 39: Evaluate $\text{arctan}(0)$ .

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

Flashcard 40: Find $\text{arccsc}(\text{√2})$ .

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

Flashcard 41: Evaluate $\text{arcsec}(2)$ .

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

Flashcard 42: 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 43: Evaluate $\text{arctan}(1)$ .

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

Flashcard 44: Identify the principal value of $\text{arccos}(-1)$ .

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

Flashcard 45: 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 46: Evaluate $\text{arctan}(1)$ .

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

Flashcard 47: What is the value of $\text{arccos}(0)$ ?

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

Flashcard 48: Evaluate $\text{arcsec}(-2)$ .

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

Flashcard 49: Find $\text{arcsin}(\frac{\text{√3}}{2})$ .

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

Flashcard 50: Evaluate $\text{arctan}(0)$ .

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

Flashcard 51: What is the inverse of $\text{sin}(\theta) = x$ ?

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

Flashcard 52: 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 53: Find $\text{arcsin}(-\frac{\text{√2}}{2})$ .

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

Flashcard 54: What is the domain of the inverse cotangent function, $\text{arccot}(x)$ ?

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

Flashcard 55: Identify the principal value of $\text{arcsin}(-1)$ .

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

Flashcard 56: 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 57: What is the principal value of $\text{arccsc}(-1)$ ?

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

Flashcard 58: 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 59: What is the principal value of $\text{arcsec}(-1)$ ?

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

Flashcard 60: What is the domain of the inverse cotangent function, $\text{arccot}(x)$ ?

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

Flashcard 61: What is the principal value of $\text{arcsec}(-1)$ ?

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

Flashcard 62: What is the principal value of $\text{arccsc}(1)$ ?

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

Flashcard 63: Evaluate $\text{arccsc}(-2)$ .

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

Flashcard 64: What is the inverse of $\text{cos}(\theta) = x$ ?

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

Flashcard 65: What is $\text{arctan}(\text{√3})$ ?

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

Flashcard 66: Find $\text{arccot}(\text{√3})$ .

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

Flashcard 67: What is the principal value of $\text{arccsc}(-1)$ ?

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

Flashcard 68: What is $\text{arcsin}(\frac{1}{2})$ ?

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

Flashcard 69: Find $\text{arcsin}(-\frac{1}{2})$ .

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

Flashcard 70: 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 71: Find $\text{arcsin}(-\frac{1}{2})$ .

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

AP Precalculus Flashcards: Inverse Trigonometric Functions

Inverse Trigonometric Functions

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: What is the inverse of sin(θ)=x\text{sin}(\theta) = xsin(θ)=x?

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

Flashcard 3: What is arccos(12)\text{arccos}(\frac{1}{2})arccos(21​)?

Flashcard 4: Find arccot(√3)\text{arccot}(\text{√3})arccot(√3).

Flashcard 5: What is arctan(√3)\text{arctan}(\text{√3})arctan(√3)?

Flashcard 6: What is the inverse of tan(θ)=x\text{tan}(\theta) = xtan(θ)=x?

Flashcard 7: What is arcsec(1)\text{arcsec}(1)arcsec(1)?

Flashcard 8: Find the value of arctan(−1)\text{arctan}(-1)arctan(−1).

Flashcard 9: Evaluate arccos(−12)\text{arccos}(-\frac{1}{2})arccos(−21​).

Flashcard 10: Find arcsin(−√22)\text{arcsin}(-\frac{\text{√2}}{2})arcsin(−2√2​).

Flashcard 11: What is the inverse of tan(θ)=x\text{tan}(\theta) = xtan(θ)=x?

Flashcard 12: Evaluate arccot(−1)\text{arccot}(-1)arccot(−1).

Flashcard 13: What is the range of the inverse cosecant function, arccsc(x)\text{arccsc}(x)arccsc(x)?

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

Flashcard 15: Evaluate arccos(1)\text{arccos}(1)arccos(1).

Flashcard 16: What is arccos(12)\text{arccos}(\frac{1}{2})arccos(21​)?

Flashcard 17: Find the value of arcsin(0)\text{arcsin}(0)arcsin(0).

Flashcard 18: Find arccsc(√2)\text{arccsc}(\text{√2})arccsc(√2).

Flashcard 19: What is the inverse of cos(θ)=x\text{cos}(\theta) = xcos(θ)=x?

Flashcard 20: Find arccos(−√32)\text{arccos}(-\frac{\text{√3}}{2})arccos(−2√3​).

Flashcard 21: What is arccot(0)\text{arccot}(0)arccot(0)?

Flashcard 22: Evaluate arccos(1)\text{arccos}(1)arccos(1).

Flashcard 23: Evaluate arccot(−1)\text{arccot}(-1)arccot(−1).

Flashcard 24: What is the domain of the inverse secant function, arcsec(x)\text{arcsec}(x)arcsec(x)?

Flashcard 25: What is the domain of the inverse cosine function, arccos(x)\text{arccos}(x)arccos(x)?

Flashcard 26: Identify the principal value of arccos(−1)\text{arccos}(-1)arccos(−1).

Flashcard 27: Identify the principal value of arcsin(−1)\text{arcsin}(-1)arcsin(−1).

Flashcard 28: Evaluate arcsec(2)\text{arcsec}(2)arcsec(2).

Flashcard 29: What is the principal value of arccsc(1)\text{arccsc}(1)arccsc(1)?

Flashcard 30: What is arcsec(1)\text{arcsec}(1)arcsec(1)?

Flashcard 31: Evaluate arcsec(−2)\text{arcsec}(-2)arcsec(−2).

Flashcard 32: What is the range of the inverse cosecant function, arccsc(x)\text{arccsc}(x)arccsc(x)?

Flashcard 33: What is arccot(0)\text{arccot}(0)arccot(0)?

Flashcard 34: What is the value of arccos(0)\text{arccos}(0)arccos(0)?

Flashcard 35: What is the domain of the inverse secant function, arcsec(x)\text{arcsec}(x)arcsec(x)?

Flashcard 36: Find the value of arctan(−1)\text{arctan}(-1)arctan(−1).

Flashcard 37: Find the value of arcsin(0)\text{arcsin}(0)arcsin(0).

Flashcard 38: Evaluate arccsc(−2)\text{arccsc}(-2)arccsc(−2).

Flashcard 39: Evaluate arctan(0)\text{arctan}(0)arctan(0).

Flashcard 40: Find arccsc(√2)\text{arccsc}(\text{√2})arccsc(√2).

Flashcard 41: Evaluate arcsec(2)\text{arcsec}(2)arcsec(2).

Flashcard 42: What is the range of the inverse tangent function, arctan(x)\text{arctan}(x)arctan(x)?

Flashcard 43: Evaluate arctan(1)\text{arctan}(1)arctan(1).

Flashcard 44: Identify the principal value of arccos(−1)\text{arccos}(-1)arccos(−1).

Flashcard 45: What is the range of the inverse cotangent function, arccot(x)\text{arccot}(x)arccot(x)?

Flashcard 46: Evaluate arctan(1)\text{arctan}(1)arctan(1).

Flashcard 47: What is the value of arccos(0)\text{arccos}(0)arccos(0)?

Flashcard 48: Evaluate arcsec(−2)\text{arcsec}(-2)arcsec(−2).

Flashcard 49: Find arcsin(√32)\text{arcsin}(\frac{\text{√3}}{2})arcsin(2√3​).

Flashcard 50: Evaluate arctan(0)\text{arctan}(0)arctan(0).

Flashcard 51: What is the inverse of sin(θ)=x\text{sin}(\theta) = xsin(θ)=x?

Flashcard 52: What is the range of the inverse cotangent function, arccot(x)\text{arccot}(x)arccot(x)?

Flashcard 53: Find arcsin(−√22)\text{arcsin}(-\frac{\text{√2}}{2})arcsin(−2√2​).

Flashcard 54: What is the domain of the inverse cotangent function, arccot(x)\text{arccot}(x)arccot(x)?

Flashcard 55: Identify the principal value of arcsin(−1)\text{arcsin}(-1)arcsin(−1).

Flashcard 56: Find arccos(−32)\text{arccos}(-\frac{\sqrt{3}}{2})arccos(−23​​).

Flashcard 57: What is the principal value of arccsc(−1)\text{arccsc}(-1)arccsc(−1)?

Flashcard 58: What is the range of the inverse sine function, arcsin(x)\text{arcsin}(x)arcsin(x)?

Flashcard 59: What is the principal value of arcsec(−1)\text{arcsec}(-1)arcsec(−1)?

Flashcard 60: What is the domain of the inverse cotangent function, arccot(x)\text{arccot}(x)arccot(x)?

Flashcard 61: What is the principal value of arcsec(−1)\text{arcsec}(-1)arcsec(−1)?

Flashcard 62: What is the principal value of arccsc(1)\text{arccsc}(1)arccsc(1)?

Flashcard 63: Evaluate arccsc(−2)\text{arccsc}(-2)arccsc(−2).

Flashcard 64: What is the inverse of cos(θ)=x\text{cos}(\theta) = xcos(θ)=x?

Flashcard 65: What is arctan(√3)\text{arctan}(\text{√3})arctan(√3)?

Flashcard 66: Find arccot(√3)\text{arccot}(\text{√3})arccot(√3).

Flashcard 67: What is the principal value of arccsc(−1)\text{arccsc}(-1)arccsc(−1)?

Flashcard 68: What is arcsin(12)\text{arcsin}(\frac{1}{2})arcsin(21​)?

Flashcard 69: Find arcsin(−12)\text{arcsin}(-\frac{1}{2})arcsin(−21​).

Flashcard 70: Evaluate arccos(−12)\text{arccos}(-\frac{1}{2})arccos(−21​).

Flashcard 71: Find arcsin(−12)\text{arcsin}(-\frac{1}{2})arcsin(−21​).