Test questions: Solving trig equations for exact values and general solutions

📚 Quick Study Guide

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• Unit Circle: Know your special angles (0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, etc.) and their corresponding sine, cosine, and tangent values.
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• Trigonometric Identities: Master the fundamental identities like $\sin^2(x) + \cos^2(x) = 1$, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and reciprocal identities.
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• General Solutions: If $\sin(x) = a$, then $x = \arcsin(a) + 2\pi k$ or $x = \pi - \arcsin(a) + 2\pi k$, where $k$ is an integer. Similarly, for cosine, if $\cos(x) = a$, then $x = \arccos(a) + 2\pi k$ or $x = -\arccos(a) + 2\pi k$. For tangent, if $\tan(x) = a$, then $x = \arctan(a) + \pi k$.
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• Exact Values: These are solutions expressed without using a calculator (e.g., $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$, 1, 0, -1).
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• Solving Steps:
  1. Isolate the trigonometric function.
  2. Find the principal value (using the unit circle or inverse trigonometric functions).
  3. Write the general solution, considering all possible angles in the interval $[0, 2\pi)$.

🧪 Practice Quiz

1. What is the general solution for $\sin(x) = \frac{1}{2}$?
1. $x = \frac{\pi}{6} + 2\pi k$
2. $x = \frac{\pi}{6} + 2\pi k$ or $x = \frac{5\pi}{6} + 2\pi k$
3. $x = \frac{\pi}{3} + 2\pi k$
4. $x = \frac{\pi}{3} + 2\pi k$ or $x = \frac{2\pi}{3} + 2\pi k$
2. What is the exact value of $x$ for $\cos(x) = \frac{\sqrt{3}}{2}$ in the interval $[0, \frac{\pi}{2}]$?
1. $\frac{\pi}{6}$
2. $\frac{\pi}{4}$
3. $\frac{\pi}{3}$
4. $\frac{\pi}{2}$
3. Find the general solution for $\tan(x) = 1$.
1. $x = \frac{\pi}{4} + \pi k$
2. $x = \frac{\pi}{4} + 2\pi k$
3. $x = \frac{5\pi}{4} + \pi k$
4. $x = \frac{5\pi}{4} + 2\pi k$
4. Solve for $x$ in the interval $[0, 2\pi)$: $2\sin(x) + 1 = 0$.
1. $\frac{\pi}{6}, \frac{5\pi}{6}$
2. $\frac{7\pi}{6}, \frac{11\pi}{6}$
3. $\frac{\pi}{3}, \frac{2\pi}{3}$
4. $\frac{4\pi}{3}, \frac{5\pi}{3}$
5. What is the general solution for $\cos(2x) = 0$?
1. $x = \frac{\pi}{4} + \pi k$
2. $x = \frac{\pi}{2} + \pi k$
3. $x = \frac{\pi}{4} + \frac{\pi}{2} k$
4. $x = \frac{\pi}{2} + 2\pi k$
6. Find the exact values of $x$ for $\sin^2(x) = 1$ in the interval $[0, 2\pi)$.
1. $0, \pi$
2. $\frac{\pi}{2}, \frac{3\pi}{2}$
3. $\frac{\pi}{4}, \frac{3\pi}{4}$
4. $\frac{\pi}{6}, \frac{5\pi}{6}$
7. Determine the general solution for $\sqrt{3}\tan(x) - 1 = 0$.
1. $x = \frac{\pi}{6} + \pi k$
2. $x = \frac{\pi}{6} + 2\pi k$
3. $x = \frac{\pi}{3} + \pi k$
4. $x = \frac{\pi}{3} + 2\pi k$