Solving Trigonometric Equations in Context - Precalculus

1. What is the principal range of $\arcsin(x)$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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