To find the period of the function \( y = \sin(2x) + 2\sin(6x) \), we need to analyze the periods of each individual sine term separately and then determine the least common period (LCP) of the entire function.

### Step 1: Find the period of \( \sin(2x) \)
The general period of \( \sin(kx) \) is \( \frac{2\pi}{k} \). For \( \sin(2x) \), we have:
\[
\text{Period of } \sin(2x) = \frac{2\pi}{2} = \pi
\]

### Step 2: Find the period of \( \sin(6x) \)
Similarly, for \( \sin(6x) \), we have:
\[
\text{Period of } \sin(6x) = \frac{2\pi}{6} = \frac{\pi}{3}
\]

### Step 3: Find the least common period
To find the overall period of the function, we take the least common multiple (LCM) of the periods \( \pi \) and \( \frac{\pi}{3} \).

The LCM of \( \pi \) and \( \frac{\pi}{3} \) is \( \pi \), since:
\[
\text{LCM}(\pi, \frac{\pi}{3}) = \pi
\]

### Conclusion:
The period of \( y = \sin(2x) + 2\sin(6x) \) is \( \pi \).

Would you like further details or have any questions?

---

Here are 5 related questions to explore:
1. How do you find the period of functions like \( \sin(kx) \) in general?
2. What would the period be if the function was \( \cos(2x) + \sin(4x) \)?
3. How does a coefficient inside the sine function affect its period?
4. Can you find the period of \( \sin^2(2x) \)?
5. What happens if we add a phase shift to the function \( \sin(2x) \)?

**Tip:** The period of a sum of sine functions is determined by the least common multiple of their individual periods.

Discover how to find the period of the trigonometric function y = sin(2x) + 2sin(6x) by analyzing the periods of the individual sine functions and calculating the least common multiple (LCM) of their periods. This problem involves key trigonometric concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation