Precalculus Flashcards: Solving Trigonometric Equations In Context

What this deck covers

This deck focuses on Solving Trigonometric Equations In Context, giving you a quick way to review the definitions, rules, and examples that matter most for 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 principal range of $\arcsin(x)$?

Answer: $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Arcsin outputs angles from $-90°$ to $90°$ (quadrants I and IV).

Flashcard 2: What is the general solution to $\sin(x)=\sin(\theta)$?

Answer: $x=\theta+2\pi k$ or $x=\pi-\theta+2\pi k$. Sine repeats every $2\pi$ with supplementary angle symmetry.

Flashcard 3: Find all solutions on $[0,2\pi)$: $\cos(x)=-\frac{\sqrt{2}}{2}$.

Answer: $x=\frac{3\pi}{4},\frac{5\pi}{4}$. Negative cosine occurs in quadrants II and III.

Flashcard 4: Identify the inverse-trig step: Solve $\sin(x)=0.5$ for the principal value of $x$.

Answer: $x=\arcsin(0.5)=\frac{\pi}{6}$. Apply arcsin to isolate $x$ in the principal range.

Flashcard 5: Find all solutions on $[0,2\pi)$: $\tan(x)=-\sqrt{3}$.

Answer: $x=\frac{2\pi}{3},\frac{5\pi}{3}$. Negative tangent occurs in quadrants II and IV.

Flashcard 6: Identify the inverse-trig step to isolate $x$ in $A\sin(Bx+C)+D=y$.

Answer: $x=\frac{\arcsin\left(\frac{y-D}{A}\right)-C}{B}$ (principal value). Subtract $D$, divide by $A$, apply arcsin, then solve for $x$.

Flashcard 7: What is the general solution to $\tan(x)=\tan(\theta)$?

Answer: $x=\theta+\pi k$. Tangent repeats every $\pi$ radians.

Flashcard 8: What domain restriction must be checked before using $\arcsin$ or $\arccos$ in an equation?

Answer: $-1\le x\le 1$. Sine and cosine only output values between $-1$ and $1$.

Flashcard 9: What is the general solution to $\cos(x)=\cos(\theta)$?

Answer: $x=\theta+2\pi k$ or $x=-\theta+2\pi k$. Cosine repeats every $2\pi$ with even function symmetry.

Flashcard 10: What is the midline of $y=A\sin(Bx+C)+D$ (or $A\cos(Bx+C)+D$)?

Answer: $y=D$. The vertical shift $D$ determines the horizontal midline.

Flashcard 11: What general solution form solves $\cos(\theta)=\cos(\alpha)$?

Answer: $\theta=\pm\alpha+2\pi k$. Cosine has same value at $\pm\alpha$, plus periodic repeats every $2\pi$.

Flashcard 12: Identify the inverse-trig step: Solve $\cos(x)=\frac{1}{2}$ for the principal value of $x$.

Answer: $x=\arccos\left(\frac{1}{2}\right)=\frac{\pi}{3}$. Apply arccos to find the principal angle.

Flashcard 13: What is the amplitude of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?

Answer: $|A|$. Amplitude is the absolute value of the coefficient $A$.

Flashcard 14: Identify the correct solutions in $[0,2\pi)$ if $\arcsin(0.8)=\alpha$ is used for $\sin(\theta)=0.8$.

Answer: $\theta=\alpha$ and $\theta=\pi-\alpha$. Arcsin gives acute angle; also need supplementary angle $\pi-\alpha$.

Flashcard 15: What is the principal range of $\arctan(x)$?

Answer: $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Arctan outputs angles strictly between $-90°$ and $90°$.

Flashcard 16: What are all solutions in $[0,2\pi)$ to $\sin(\theta)=\frac{1}{2}$?

Answer: $\theta=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$; sine positive in quadrants I and II.

Flashcard 17: What is the vertical-range check for real solutions in $A\cos(\cdot)+D=y$?

Answer: $D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.

Flashcard 18: Solve on $[0,2\pi)$: $2\sin(x)-1=0$.

Answer: $x=\frac{\pi}{6},\frac{5\pi}{6}$. First isolate $\sin(x)=\frac{1}{2}$, then find all angles.

Flashcard 19: What is the period of $y=A\sin(Bx+C)+D$ (or $A\cos(Bx+C)+D$) in radians?

Answer: $\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of frequency $B$.

Flashcard 20: Find all solutions on $[0,2\pi)$: $\sin(x)=\frac{1}{2}$.

Answer: $x=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$ occurs in quadrants I and II.

Flashcard 21: What is the domain of $\arctan(x)$?

Answer: $x\in(-\infty,\infty)$. Tangent can take any real value as input.

Flashcard 22: Interpret feasibility: If a model gives $\sin(\theta)=1.2$, what conclusion is correct?

Answer: No real solution because $1.2\notin\left[-1,1\right]$. Sine values must be in $[-1,1]$ for real solutions.

Flashcard 23: What are all solutions in $[0,2\pi)$ to $\tan(\theta)=\sqrt{3}$?

Answer: $\theta=\frac{\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; tangent positive in quadrants I and III.

Flashcard 24: What is the vertical-range check for real solutions in $A\sin(\cdot)+D=y$?

Answer: $D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.

Flashcard 25: What general solution form solves $\tan(\theta)=\tan(\alpha)$?

Answer: $\theta=\alpha+\pi k$. Tangent repeats every $\pi$ radians (half the period of sine/cosine).

Flashcard 26: What is the period of $y=A\tan(Bx+C)+D$?

Answer: $\frac{\pi}{|B|}$. Tangent's period is half that of sine/cosine with same $B$.

Flashcard 27: Identify the inverse-trig step to isolate $x$ in $A\cos(Bx+C)+D=y$.

Answer: $x=\frac{\arccos\left(\frac{y-D}{A}\right)-C}{B}$ (principal value). Subtract $D$, divide by $A$, apply arccos, then solve for $x$.

Flashcard 28: What contextual filtering is required after solving a trig model for time $t$?

Answer: Keep only $t$ values in the stated interval and with correct units. Discard solutions outside domain or with wrong units for the context.

Flashcard 29: What is the principal range of $\arccos(x)$?

Answer: $\left[0,\pi\right]$. Arccos outputs angles from $0°$ to $180°$ (quadrants I and II).

Flashcard 30: Identify the inverse-trig step: Solve $\tan(x)=1$ for the principal value of $x$.

Answer: $x=\arctan(1)=\frac{\pi}{4}$. Apply arctan to find the principal angle.

Flashcard 31: What general solution form solves $\sin(\theta)=\sin(\alpha)$?

Answer: $\theta=\alpha+2\pi k$ or $\theta=\pi-\alpha+2\pi k$. Sine has same value at supplementary angles, plus periodic repeats.

Flashcard 32: What is the domain of $\arcsin(x)$ and $\arccos(x)$?

Answer: $x\in\left[-1,1\right]$. Sine and cosine values range from $-1$ to $1$.

Flashcard 33: What are all solutions in $[0,2\pi)$ to $\cos(\theta)=-\frac{1}{2}$?

Answer: $\theta=\frac{2\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; cosine negative in quadrants II and III.

Flashcard 34: What is the period of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?

Answer: $\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of $B$.

Flashcard 35: What is the amplitude of $y=A\sin(Bx+C)+D$ (or $A\cos(Bx+C)+D$)?

Answer: $|A|$. Amplitude is the absolute value of the coefficient $A$.