Grade 11 Sine and Cosine Equations Worksheet
Solve and analyze sine and cosine equations, focusing on general solutions and solutions within specific intervals.
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Grade 11 Sine and Cosine Equations
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Read each question carefully and solve for the unknown variable. Show all your work for full credit.
1. Find all general solutions for the equation: 2 sin(x) - 1 = 0
2. Find all general solutions for the equation: cos(x) + \(\sqrt{2}\) = 0
3. The solutions to \( \sin(x) = \frac{1}{2} \) in the interval \([0, 2\pi)\) are x = and x = .
4. The solutions to \( \cos(x) = -\frac{\sqrt{3}}{2} \) in the interval \([0, 2\pi)\) are x = and x = .
5. Which of the following is a general solution for \( \tan(x) = 1 \)?
\( x = \frac{\pi}{4} + 2n\pi \)
\( x = \frac{3\pi}{4} + n\pi \)
\( x = \frac{\pi}{4} + n\pi \)
\( x = \frac{5\pi}{4} + 2n\pi \)
6. How many solutions does \( \cos(x) = 0.5 \) have in the interval \([-\pi, \pi]\)?
1
2
3
4
7. Find all general solutions for the equation: \( 2\cos^2(x) - \cos(x) - 1 = 0 \)
8. Solve \( \sin(2x) = \frac{\sqrt{3}}{2} \) for \( x \in [0, 2\pi) \).
9. The equation \( \sin(x) = 2 \) has no real solutions.
True
False
10. All solutions to \( \cos(x) = 1 \) are of the form \( x = 2n\pi \) where \( n \) is an integer.
True
False
11. Consider the equation \( 3\sin(x) - \cos(x) = 0 \). Find all general solutions.