Solved Examples: Trigonometric Equations with Multiple Angles (Pre-Calculus)

📚 Quick Study Guide

• 📐 Trigonometric Identities: Remember your basic trig identities like $\sin^2(x) + \cos^2(x) = 1$, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and their variations.
• 👯 Double Angle Formulas: These are crucial! $\sin(2x) = 2\sin(x)\cos(x)$, $\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1$, and $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$.
• 🧑‍ triple Triple Angle Formulas: Although less common, know them. $\sin(3x) = 3\sin(x) - 4\sin^3(x)$, $\cos(3x) = 4\cos^3(x) - 3\cos(x)$.
• ➕ Sum and Difference Formulas: $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$, $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$.
• 💡 General Solutions: When solving, remember the periodic nature of trigonometric functions. For example, if $\sin(x) = a$, then $x = \arcsin(a) + 2\pi k$ or $x = \pi - \arcsin(a) + 2\pi k$, where $k$ is an integer. Don't forget to account for all possible solutions within the desired interval!
• 🔄 Substitution: Sometimes substituting $u$ for the multiple angle (e.g., $u = 2x$) can simplify the equation. Solve for $u$ first, then solve for $x$.

✍️ Practice Quiz

1. What is the general solution to the equation $\sin(2x) = 0.5$?
1. A) $x = \frac{\pi}{12} + k\pi$ or $x = \frac{5\pi}{12} + k\pi$, where $k$ is an integer.
2. B) $x = \frac{\pi}{6} + 2k\pi$ or $x = \frac{5\pi}{6} + 2k\pi$, where $k$ is an integer.
3. C) $x = \frac{\pi}{3} + k\pi$ or $x = \frac{2\pi}{3} + k\pi$, where $k$ is an integer.
4. D) $x = \frac{\pi}{12} + 2k\pi$ or $x = \frac{5\pi}{12} + 2k\pi$, where $k$ is an integer.
2. Solve for $x$ in the interval $[0, \pi]$: $\cos(2x) = \cos(x)$
1. A) $0, \frac{2\pi}{3}, \pi$
2. B) $\frac{\pi}{3}, \pi$
3. C) $0, \frac{\pi}{2}, \pi$
4. D) $0, \frac{\pi}{4}, \frac{\pi}{2}$
3. Simplify the equation $\sin(3x) + \sin(x) = 0$. Which of the following is a possible solution?
1. A) $x = \frac{\pi}{8}$
2. B) $x = \frac{\pi}{4}$
3. C) $x = \frac{\pi}{3}$
4. D) $x = \frac{\pi}{2}$
4. Find the solutions to $\tan(2x) = 1$ in the interval $[0, \pi]$.
1. A) $\frac{\pi}{8}, \frac{5\pi}{8}$
2. B) $\frac{\pi}{4}, \frac{3\pi}{4}$
3. C) $\frac{\pi}{2}, \pi$
4. D) $\frac{\pi}{12}, \frac{7\pi}{12}$
5. Solve $\cos(4x) = 0$ for $x$ in $[0, \frac{\pi}{2}]$.
1. A) $\frac{\pi}{8}, \frac{3\pi}{8}$
2. B) $\frac{\pi}{16}, \frac{3\pi}{16}, \frac{5\pi}{16}, \frac{7\pi}{16}$
3. C) $\frac{\pi}{4}, \frac{3\pi}{4}$
4. D) $\frac{\pi}{2}, \pi$
6. What is the general solution for $\sin(3x) = \sin(x)$?
1. A) $k\pi$ or $\frac{\pi}{4} + k\frac{\pi}{2}$, where $k$ is an integer.
2. B) $2k\pi$ or $\frac{\pi}{2} + k\pi$, where $k$ is an integer.
3. C) $k\frac{\pi}{2}$ or $\frac{\pi}{3} + k\frac{\pi}{2}$, where $k$ is an integer.
4. D) $k\pi$ or $\frac{\pi}{2} + k\pi$, where $k$ is an integer.
7. If $\cos(2x) + \sin(x) = 0$, find the solutions for $x$ in $[0, 2\pi]$.
1. A) $\frac{7\pi}{6}, \frac{11\pi}{6}, \frac{\pi}{2}$
2. B) $\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$
3. C) $\frac{7\pi}{6}, \frac{11\pi}{6}, \frac{3\pi}{2}$
4. D) $\frac{\pi}{3}, \frac{2\pi}{3}, \frac{\pi}{2}$