Determining the period of a trigonometric function

In summary, the period of f(x) is the lowest common multiple of the periods of sin2x and cos4x, which is π. This is determined by finding a common multiple of the periods, n*p_1=m*p_2, where n and m are integers. The fact that cos4x is a displacement of sin2x on the x-axis does not affect the period, as both functions repeat themselves at a common period of π.
  • #1
ainster31
158
1

Homework Statement



$$f(x)=sin2x+cos4x$$

Homework Equations




The Attempt at a Solution



$$The\quad period\quad of\quad sin2x\quad is\quad π.\quad The\quad period\quad of\quad cos4x\quad is\quad \frac { π }{ 2 } .\\ \\ What\quad is\quad the\quad period\quad of\quad f(x)?$$
 
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  • #2
ainster31 said:

Homework Statement



$$f(x)=sin2x+cos4x$$

Homework Equations




The Attempt at a Solution



$$The\quad period\quad of\quad sin2x\quad is\quad π.\quad The\quad period\quad of\quad cos4x\quad is\quad \frac { π }{ 2 } .\\ \\ What\quad is\quad the\quad period\quad of\quad f(x)?$$

What is the lowest common multiple of the periods of sin2x and cos4x ?
 
  • #3
Suppose you have two periods, p_1 and p_2
the COMMON period must then satisfy n*p_1=m*p_2, for integers n and m to be determined.

That is, the common period must be, as Tanya Sharma says, a COMMON MULTIPLE of the two periods, and the LEAST one at that. :smile:
 
  • #4
How does that work considering that cos is sin but displaced on the x-axis?

According to this graph, T=pi is wrong:

eZclCyl.png
 
  • #5
Never mind.

I'm an idiot. You're right.

Thanks.
 
  • #6
No,it isn't.

Look at the respective function values of sin(2x) and cos(4x) at two x's a pi apart.

BOTH functions repeat THEMSELVES here, whether you use -pi/3 and 2pi/3 or "0 and pi" or whatever other couple of x's.

The common period of two functions does NOT mean that they equal each other at those points.
 

1. How do you determine the period of a trigonometric function?

The period of a trigonometric function is the length of one complete cycle of the function. To determine the period, you can look at the coefficient of the variable inside the trigonometric function. For example, in the function y = sin(2x), the coefficient of x is 2. The period of this function would be 2π, since the function completes one full cycle in that interval.

2. Can the period of a trigonometric function be negative?

No, the period of a trigonometric function cannot be negative. The period represents a length of time or distance, which cannot be negative. It is always a positive value.

3. Can you determine the period of a trigonometric function without a graph?

Yes, you can determine the period of a trigonometric function without a graph by looking at the equation and identifying the coefficient of the variable inside the function. The period will be equal to 2π divided by the coefficient.

4. How does the amplitude of a trigonometric function affect the period?

The amplitude of a trigonometric function does not affect the period. The amplitude only affects the vertical height of the function, while the period determines the length of one cycle. The two are independent of each other.

5. Can the period of a trigonometric function change?

Yes, the period of a trigonometric function can change depending on the value of the coefficient of the variable inside the function. A larger coefficient will result in a shorter period, while a smaller coefficient will result in a longer period.

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