Find the period of the f(x)=sin10x + cos3x

To find the period of the function \( f(x) = \sin(10x) + \cos(3x) \), we need to determine the individual periods of both trigonometric components and then find their least common multiple (LCM).

### Step 1: Find the period of \( \sin(10x) \)
The general period of \( \sin(kx) \) is given by:

\[
T_{\sin} = \frac{2\pi}{k}
\]

For \( \sin(10x) \), the period is:

\[
T_{\sin} = \frac{2\pi}{10} = \frac{\pi}{5}
\]

### Step 2: Find the period of \( \cos(3x) \)
Similarly, the general period of \( \cos(kx) \) is:

\[
T_{\cos} = \frac{2\pi}{k}
\]

For \( \cos(3x) \), the period is:

\[
T_{\cos} = \frac{2\pi}{3}
\]

### Step 3: Find the least common multiple (LCM)
To find the overall period of the function, we need to calculate the LCM of the individual periods:

\[
\text{LCM}\left(\frac{\pi}{5}, \frac{2\pi}{3}\right)
\]

The LCM is found by taking the least common multiple of the denominators (5 and 3), which is 15. So, the overall period is:

\[
T = \frac{2\pi}{15}
\]

### Conclusion:
The period of the function \( f(x) = \sin(10x) + \cos(3x) \) is \( \frac{2\pi}{15} \).

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What is the period of \( \sin(5x) + \cos(2x) \)?
2. How do you determine the period of a trigonometric function in general?
3. What is the period of \( f(x) = \sin(6x) + \cos(4x) \)?
4. How can the LCM of fractions be computed efficiently?
5. How does changing the coefficients in front of \(x\) affect the period of sine and cosine functions?

**Tip**: When combining trigonometric functions, always find the LCM of the individual periods to get the period of the combined function.

Learn how to find the period of the function f(x) = sin(10x) + cos(3x) by determining the periods of both sine and cosine functions and calculating the least common multiple. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation