AP Precalculus Flashcards: Trigonometric Equations And Inequalities

What this deck covers

This deck focuses on Trigonometric Equations And Inequalities, 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: Solve $\text{tan}(\theta) = \frac{1}{\sqrt{3}}$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{6}, \frac{7\pi}{6}$. Since $\tan(\theta) = \frac{1}{\sqrt{3}}$, we have $\theta = \frac{\pi}{6}$ and $\frac{7\pi}{6}$.

Flashcard 2: Find the solutions for $\text{sin}(\theta) = -\frac{\text{√2}}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Sine equals $-\frac{\sqrt{2}}{2}$ in the third and fourth quadrants.

Flashcard 3: Find the solutions for $\cot(\theta) = 0$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{2}, \frac{3\pi}{2}$. Cotangent equals zero when cosine is zero.

Flashcard 4: State the double angle formula for $\text{cos}(2\theta)$.

Answer: $\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$. This is the standard double angle formula for cosine.

Flashcard 5: Solve $\text{sin}(2\theta) = 0$ for $\theta$ in $[0, 2\text{π})$.

Answer: $\theta = 0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2}$. When $\sin(2\theta) = 0$, then $2\theta = n\text{π}$, so $\theta = \frac{n\text{π}}{2}$.

Flashcard 6: What is the period of $\text{cos}(2\theta)$?

Answer: $\text{π}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

Flashcard 7: What is the general solution for $\text{csc}(\theta) = a$?

Answer: $\theta = \text{csc}^{-1}(a) + 2n\text{π} \text{ or } -\text{csc}^{-1}(a) + 2n\text{π}$. Cosecant is symmetric, giving two solutions per period.

Flashcard 8: Find the solutions for $\text{cot}(\theta) = 0$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{2}, \frac{3\text{π}}{2}$. Cotangent equals zero when cosine is zero.

Flashcard 9: Solve $\text{cos}^2(\theta) = \frac{1}{2}$ for $\theta$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{4}, \frac{3\text{π}}{4}, \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Taking square root gives $\cos(\theta) = \pm\frac{\sqrt{2}}{2}$, yielding four solutions.

Flashcard 10: State the identity for $\text{sec}(\theta)$ in terms of $\text{cos}(\theta)$.

Answer: $\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}$. Secant is the reciprocal of cosine.

Flashcard 11: What is the general solution for $\text{sin}(\theta) = 0$?

Answer: $\theta = n\text{π}$, $n \text{ is an integer}$. Sine equals zero at multiples of $\text{π}$.

Flashcard 12: What is the general solution for $\text{sec}(\theta) = a$?

Answer: $\theta = \text{sec}^{-1}(a) + 2n\text{π} \text{ or } -\text{sec}^{-1}(a) + 2n\text{π}$. Secant is symmetric about the y-axis, giving two solutions per period.

Flashcard 13: Find the solutions for $\text{sin}(\theta) = 1$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{2}$. Sine equals 1 only at $\frac{\text{π}}{2}$ in the interval $[0, 2\text{π})$.

Flashcard 14: Find the solutions for $\text{sin}(\theta) = 1$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{2}$. Sine equals 1 only at $\frac{\text{π}}{2}$ in the interval $[0, 2\text{π})$.

Flashcard 15: What is the period of $\text{tan}(4\theta)$?

Answer: $\frac{\text{π}}{4}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{4}$ for $b=4$.

Flashcard 16: What is the period of $\tan(3\theta)$?

Answer: $\frac{\text{π}}{3}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

Flashcard 17: What is the general solution for $\text{cos}(\theta) = a$?

Answer: $\theta = \text{cos}^{-1}(a) + 2n\text{π} \text{ or } -\text{cos}^{-1}(a) + 2n\text{π}$. Cosine is symmetric about the y-axis, giving two solutions per period.

Flashcard 18: What is the period of $\text{csc}(\theta)$?

Answer: $2\text{π}$. Cosecant has the same period as sine, which is $2\text{π}$.

Flashcard 19: What is the period of $\text{cot}(3\theta)$?

Answer: $\frac{\text{π}}{3}$. Period of $\cot(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

Flashcard 20: Find the solutions for $\text{cos}(\theta) = -\frac{1}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Cosine equals $-\frac{1}{2}$ at angles $\frac{2\text{π}}{3}$ and $\frac{4\text{π}}{3}$.

Flashcard 21: Find $\theta$ if $\text{sec}(\theta) = 2$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{3}, \frac{5\text{π}}{3}$. Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we need $\cos(\theta) = \frac{1}{2}$.

Flashcard 22: State the identity for $\text{csc}(\theta)$ in terms of $\text{sin}(\theta)$.

Answer: $\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}$. Cosecant is the reciprocal of sine.

Flashcard 23: Solve $\text{cos}(\theta) = \frac{1}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{3}, \frac{5\text{π}}{3}$. Cosine equals $\frac{1}{2}$ at angles $\frac{\text{π}}{3}$ and $\frac{5\text{π}}{3}$ in $[0, 2\text{π})$.

Flashcard 24: What is the period of $\text{cos}(4\theta)$?

Answer: $\frac{\text{π}}{2}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

Flashcard 25: Solve $\tan(\theta) = \frac{1}{\sqrt{3}}$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{6}, \frac{7\pi}{6}$. Since $\tan(\theta) = \frac{1}{\sqrt{3}}$, we have $\theta = \frac{\pi}{6}$ and $\frac{7\pi}{6}$.

Flashcard 26: Find the solutions for $\text{sin}(\theta) = -\frac{\text{√2}}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Sine equals $-\frac{\sqrt{2}}{2}$ in the third and fourth quadrants.

Flashcard 27: What is the period of $\text{cot}(\theta)$?

Answer: $\text{π}$. Cotangent function repeats its values every $\text{π}$ radians.

Flashcard 28: What is the period of $\text{csc}(2\theta)$?

Answer: $\text{π}$. Period of $\csc(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

Flashcard 29: Solve $\text{cos}^2(\theta) = \frac{1}{2}$ for $\theta$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{4}, \frac{3\text{π}}{4}, \frac{5\text{π}}{4}, \frac{7\text{π}}{4}$. Taking square root gives $\cos(\theta) = \pm\frac{\sqrt{2}}{2}$, yielding four solutions.

Flashcard 30: What is the general solution for $\text{sin}(\theta) = a$?

Answer: $\theta = \text{sin}^{-1}(a) + 2n\text{π} \text{ or } \text{π} - \text{sin}^{-1}(a) + 2n\text{π}$. Sine has two solutions per period due to its symmetry properties.

Flashcard 31: What is the period of $\text{sin}(4\theta)$?

Answer: $\frac{\text{π}}{2}$. Period of $\sin(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

Flashcard 32: Find the solutions for $\text{cos}(\theta) = 0$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{2}, \frac{3\text{π}}{2}$. Cosine equals zero when the angle is an odd multiple of $\frac{\text{π}}{2}$.

Flashcard 33: What is the period of $\text{sin}(\theta)$?

Answer: $2\text{π}$. Sine function completes one cycle every $2\text{π}$ radians.

Flashcard 34: What is the period of $\text{sec}(\theta)$?

Answer: $2\text{π}$. Secant has the same period as cosine, which is $2\text{π}$.

Flashcard 35: Solve $\text{sin}(2\theta) = 0$ for $\theta$ in $[0, 2\text{π})$.

Answer: $\theta = 0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2}$. When $\sin(2\theta) = 0$, then $2\theta = n\text{π}$, so $\theta = \frac{n\text{π}}{2}$.

Flashcard 36: What is the period of $\tan(\theta)$?

Answer: $\text{π}$. Tangent function repeats its values every $\text{π}$ radians.

Flashcard 37: State the identity for $\text{csc}(\theta)$ in terms of $\text{sin}(\theta)$.

Answer: $\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}$. Cosecant is the reciprocal of sine.

Flashcard 38: Solve $\text{cos}(\theta) = \frac{1}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{3}, \frac{5\text{π}}{3}$. Cosine equals $\frac{1}{2}$ at angles $\frac{\text{π}}{3}$ and $\frac{5\text{π}}{3}$ in $[0, 2\text{π})$.

Flashcard 39: Solve $\sin(\theta) = \frac{\sqrt{3}}{2}$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{3}, \frac{2\pi}{3}$. Sine equals $\frac{\sqrt{3}}{2}$ at angles $\frac{\pi}{3}$ and $\frac{2\pi}{3}$.

Flashcard 40: Find the solutions for $\text{csc}(\theta) = -2$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{7\text{π}}{6}, \frac{11\text{π}}{6}$. Since $\csc(\theta) = -2$, we need $\sin(\theta) = -\frac{1}{2}$.

Flashcard 41: State the Pythagorean identity for $\text{tan}^2(\theta)$.

Answer: $1 + \tan^2(\theta) = \text{sec}^2(\theta)$. This is the fundamental Pythagorean identity involving tangent and secant.

Flashcard 42: What is the period of $\text{tan}(4\theta)$?

Answer: $\frac{\text{π}}{4}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{4}$ for $b=4$.

Flashcard 43: What is the period of $\text{csc}(\theta)$?

Answer: $2\text{π}$. Cosecant has the same period as sine, which is $2\text{π}$.

Flashcard 44: Find the solutions for $\text{cos}(\theta) = -\frac{1}{2}$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Cosine equals $-\frac{1}{2}$ at angles $\frac{2\text{π}}{3}$ and $\frac{4\text{π}}{3}$.

Flashcard 45: State the Pythagorean identity for $\text{tan}^2(\theta)$.

Answer: $1 + \tan^2(\theta) = \text{sec}^2(\theta)$. This is the fundamental Pythagorean identity involving tangent and secant.

Flashcard 46: What is the period of $\text{cot}(3\theta)$?

Answer: $\frac{\text{π}}{3}$. Period of $\cot(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

Flashcard 47: What is the general solution for $\text{cos}(\theta) = a$?

Answer: $\theta = \text{cos}^{-1}(a) + 2n\text{π} \text{ or } -\text{cos}^{-1}(a) + 2n\text{π}$. Cosine is symmetric about the y-axis, giving two solutions per period.

Flashcard 48: Find the solutions for $\text{tan}(\theta) = 1$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{4}, \frac{5\text{π}}{4}$. Tangent equals 1 at $\frac{\text{π}}{4}$ and repeats every $\text{π}$.

Flashcard 49: Find the solutions for $\text{tan}(\theta) = 1$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{4}, \frac{5\text{π}}{4}$. Tangent equals 1 at $\frac{\text{π}}{4}$ and repeats every $\text{π}$.

Flashcard 50: Solve $\sin(\theta) = \frac{\sqrt{3}}{2}$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{3}, \frac{2\pi}{3}$. Sine equals $\frac{\sqrt{3}}{2}$ at angles $\frac{\pi}{3}$ and $\frac{2\pi}{3}$.

Flashcard 51: What is the general solution for $\text{csc}(\theta) = a$?

Answer: $\theta = \text{csc}^{-1}(a) + 2n\text{π} \text{ or } -\text{csc}^{-1}(a) + 2n\text{π}$. Cosecant is symmetric, giving two solutions per period.

Flashcard 52: Find the solutions for $\text{csc}(\theta) = -2$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{7\text{π}}{6}, \frac{11\text{π}}{6}$. Since $\csc(\theta) = -2$, we need $\sin(\theta) = -\frac{1}{2}$.

Flashcard 53: What is the period of $\tan(\theta)$?

Answer: $\text{π}$. Tangent function repeats its values every $\text{π}$ radians.

Flashcard 54: What is the period of $\text{cos}(4\theta)$?

Answer: $\frac{\text{π}}{2}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

Flashcard 55: What is the amplitude of $\text{sin}(3\theta)$?

Answer:

1. Amplitude is the coefficient of the sine function, which is 1.

Flashcard 56: What is the period of $\text{cot}(\theta)$?

Answer: $\text{π}$. Cotangent function repeats its values every $\text{π}$ radians.

Flashcard 57: What is the general solution for $\text{sin}(\theta) = 0$?

Answer: $\theta = n\text{π}$, $n \text{ is an integer}$. Sine equals zero at multiples of $\text{π}$.

Flashcard 58: What is the period of $\tan(3\theta)$?

Answer: $\frac{\text{π}}{3}$. Period of $\tan(b\theta)$ is $\frac{\text{π}}{b}$, so $\frac{\text{π}}{3}$ for $b=3$.

Flashcard 59: What is the amplitude of $\text{sin}(3\theta)$?

Answer:

1. Amplitude is the coefficient of the sine function, which is 1.

Flashcard 60: Find $\theta$ if $\sec(\theta) = 2$ in $[0, 2\pi)$.

Answer: $\theta = \frac{\pi}{3}, \frac{5\pi}{3}$. Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we need $\cos(\theta) = \frac{1}{2}$.

Flashcard 61: What is the period of $\text{sec}(\theta)$?

Answer: $2\text{π}$. Secant has the same period as cosine, which is $2\text{π}$.

Flashcard 62: What is the period of $\text{sin}(\theta)$?

Answer: $2\text{π}$. Sine function completes one cycle every $2\text{π}$ radians.

Flashcard 63: State the identity for $\text{sec}(\theta)$ in terms of $\text{cos}(\theta)$.

Answer: $\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}$. Secant is the reciprocal of cosine.

Flashcard 64: State the double angle formula for $\text{cos}(2\theta)$.

Answer: $\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$. This is the standard double angle formula for cosine.

Flashcard 65: What is the period of $\text{sin}(4\theta)$?

Answer: $\frac{\text{π}}{2}$. Period of $\sin(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{4} = \frac{\text{π}}{2}$.

Flashcard 66: What is the general solution for $\text{sec}(\theta) = a$?

Answer: $\theta = \text{sec}^{-1}(a) + 2n\text{π} \text{ or } -\text{sec}^{-1}(a) + 2n\text{π}$. Secant is symmetric about the y-axis, giving two solutions per period.

Flashcard 67: What is the period of $\text{cos}(2\theta)$?

Answer: $\text{π}$. Period of $\cos(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

Flashcard 68: Find the solutions for $\text{cos}(\theta) = 0$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{\text{π}}{2}, \frac{3\text{π}}{2}$. Cosine equals zero when the angle is an odd multiple of $\frac{\text{π}}{2}$.

Flashcard 69: What is the period of $\text{csc}(2\theta)$?

Answer: $\text{π}$. Period of $\csc(b\theta)$ is $\frac{2\text{π}}{b}$, so $\frac{2\text{π}}{2} = \text{π}$.

Flashcard 70: What is the general solution for $\text{sin}(\theta) = a$?

Answer: $\theta = \text{sin}^{-1}(a) + 2n\text{π} \text{ or } \text{π} - \text{sin}^{-1}(a) + 2n\text{π}$. Sine has two solutions per period due to its symmetry properties.

Flashcard 71: Find the solutions for $\text{sec}(\theta) = -2$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Since $\sec(\theta) = -2$, we need $\cos(\theta) = -\frac{1}{2}$.

Flashcard 72: Find the solutions for $\text{sec}(\theta) = -2$ in $[0, 2\text{π})$.

Answer: $\theta = \frac{2\text{π}}{3}, \frac{4\text{π}}{3}$. Since $\sec(\theta) = -2$, we need $\cos(\theta) = -\frac{1}{2}$.