what is the period of y = 2 cos x - sin 3 x

To find the period of the function \( y = 2 \cos(x) - \sin(3x) \), we need to examine the periods of both trigonometric components individually and then determine the least common period.

### 1. Period of \( \cos(x) \):
- The standard period of \( \cos(x) \) is \( 2\pi \). Since there is no coefficient affecting the \( x \) inside the cosine function, the period of \( 2 \cos(x) \) remains \( 2\pi \).

### 2. Period of \( \sin(3x) \):
- The standard period of \( \sin(x) \) is \( 2\pi \), but when \( x \) is multiplied by a factor \( 3 \), the period becomes \( \frac{2\pi}{3} \).

### Finding the overall period:
- The overall period of the function will be the least common multiple (LCM) of the two individual periods:
  - The period of \( 2 \cos(x) \) is \( 2\pi \).
  - The period of \( \sin(3x) \) is \( \frac{2\pi}{3} \).

To find the LCM of \( 2\pi \) and \( \frac{2\pi}{3} \), we determine the smallest multiple that both periods share:

\[
\text{LCM}(2\pi, \frac{2\pi}{3}) = 2\pi
\]

Thus, the period of the function \( y = 2 \cos(x) - \sin(3x) \) is \( 2\pi \).

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you find the period of a function with multiple trigonometric terms?
2. What happens to the period if we change the coefficient inside the cosine function?
3. How would the period change if the sine term was \( \sin(5x) \) instead?
4. Can we find the phase shift of a function like \( y = 2 \cos(x) - \sin(3x) \)?
5. What are the amplitudes of the cosine and sine functions in this equation?

**Tip**: When analyzing a sum of trigonometric functions, always start by finding the period of each term individually.

Learn how to find the period of the trigonometric function y = 2 cos(x) - sin(3x) by analyzing individual components. Discover how the periods of cos(x) and sin(3x) contribute to the overall period of 2π. Suitable for students in Grades 10-12.

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