📚 Topic Summary
Solving equations of the form $\sin(nx) = k$ involves finding the values of $x$ that satisfy the equation, where $n$ is a constant and $k$ is a value between -1 and 1. The general strategy involves finding the principal solutions using the inverse sine function ($\arcsin$) and then accounting for the periodic nature of the sine function to find all possible solutions within a given interval. Remember that the sine function has a period of $2\pi$, and $\sin(\theta) = \sin(\pi - \theta)$.
When solving $\sin(nx) = k$, you first find the reference angle $\alpha = \arcsin(k)$. Then, the general solutions for $nx$ are given by $nx = \alpha + 2m\pi$ and $nx = (\pi - \alpha) + 2m\pi$, where $m$ is an integer. Finally, divide by $n$ to solve for $x$. Always check your solutions to ensure they are valid!
🧮 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Sine |
A. The angle whose sine is a given number. |
| 2. Period |
B. The ratio of the length of the side opposite the angle to the length of the hypotenuse. |
| 3. Arcsine |
C. The interval after which the function repeats its values. |
| 4. Amplitude |
D. The maximum displacement from the equilibrium position. |
| 5. Radian |
E. The standard unit of angular measure, where $2\pi$ radians equals 360 degrees. |
Answers: 1-B, 2-C, 3-A, 4-D, 5-E
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
To solve $\sin(nx) = k$, first find the ______ angle $\alpha = \arcsin(k)$. The general solutions for $nx$ are $nx = \alpha + 2m\pi$ and $nx = (\pi - \alpha) + 2m\pi$, where $m$ is an _______. Divide by $n$ to solve for _______. The sine function has a _______ of $2\pi$. Always _______ your solutions.
Possible words: integer, check, reference, period, $x$
Answers: reference, integer, $x$, period, check
🤔 Part C: Critical Thinking
Explain why there might be multiple solutions to the equation $\sin(nx) = k$ within the interval $[0, 2\pi]$.