how to find period of trig function

Period of the Sine Function – Formulas and Examples

The menstruum of the sine function is 2π. This means that the value of the function is the same every 2π units. Similar to other trigonometric functions, the sine role is a periodic function, which means that it repeats at regular intervals. The interval of the sine function is 2π.

For example, nosotros have sin(π) = 0. If we add together 2π to the input of the part, we have sin(π + 2π), which is equal to sin(3π). Since we have sin(π) = 0, nosotros besides accept sin(3π) = 0. Every time we add 2π of the input values, nosotros will become the same issue.

TRIGONOMETRY

Relevant for…

Learning to discover the menstruum of the sine role.

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TRIGONOMETRY

Relevant for…

Learning to discover the flow of the sine function.

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Period of the bones sine function

The basic sine function is $y = \sin(x)$ . Since this part tin be evaluated for any real number, the sine function is defined for all real numbers. The flow of this part can be conspicuously observed in its graph since it is the distance between "equivalent" points.

Since the graph of $y = \sin(x)$ looks like a single pattern that repeats itself over and over, we tin recall of the period as the distance on the x-axis before the graph starts repeating.

diagram of the period of the sine function

Looking at the graph, we see that the graph repeats itself afterward 2π. This means that the function is periodic with a period of 2π. In the unit circle, 2π equals one consummate revolution effectually the circle.

Whatsoever quantity greater than 2π means that we are repeating the revolution. This is the reason why the value of the function is the aforementioned every 2π.

Irresolute the period of the sine office

The period of the basic sine function $y = \sin (x)$ is 2π, but ifx is multiplied by a abiding, the period of the function can change.

Ifx is multiplied by a number greater than one, that "speeds up" the part and the flow will exist smaller. That ways it won't have long for the part to start repeating itself.

For case, if we have the function $y = \sin(2x)$ , this causes the "speed" of the original function to double. In this case, the period is π.

On the other hand, ifx is multiplied past a number between 0 and 1, this causes the function to slow down and the menstruation will be larger since it will take longer for the office to repeat itself. For instance, the function $y = \sin (\frac{x}{2})$ halves the "speed" of the original role. The menstruum of this role is 4π.

Finding the period of a sine function

To find the period of a sine office, nosotros take to consider the coefficient ofx that is within the part. Nosotros can use B to represent this coefficient. Therefore, if nosotros accept an equation in the form $y = \sin(Bx)$ , we accept the post-obit formula:

$\text{Period}=\frac{2\pi}{|B|}$

In the denominator, we have |B|. This means that nosotros take the absolute value of B. Thus, if B is a negative number, we just accept the positive version of the number.

This formula works fifty-fifty if we have more complex variations of the sine function similar $y = 3 \sin(2x + 4)$ . Only the coefficient often matters when calculating the catamenia, and then we would take:

$\text{Period}=\frac{2\pi}{|B|}$

$\text{Period}=\frac{2\pi}{2}$

$\text{Period}=\pi$

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Period of the sine function – Examples with answers

What yous have learned about the menstruum of sine functions is used to solve the following examples. Try to solve the problems yourself before looking at the answer.

EXAMPLE 1

What is the period of the function $y = \sin(3x)$ ?

Solution

We use the menses formula with the value $| B | = 3$ . Therefore, we take:

$\text{Period}=\frac{2\pi}{|B|}$

$\text{Period}=\frac{2\pi}{3}$

The period of the function is $\frac{2}{3}\pi$ .

EXAMPLE 2

We have the sine office $y = 3 \sin(4x)+1$ . What is its period?

Solution

The only value we demand is the coefficient ofx. Therefore, we utilise the value $| B | = 4$ in the formula for the period:

$\text{Period}=\frac{2\pi}{|B|}$

$\text{Period}=\frac{2\pi}{4}$

$\text{Period}=\frac{\pi}{2}$

The catamenia of the function is $\frac{\pi}{2}$ .

Instance three

What is the period of the function $y = \frac{1}{2} \sin (- \frac{1}{4} x-4)$ ?

Solution

Nosotros only have to use the coefficient ofx to find the catamenia. We see that in this case, the coefficient is negative, so nosotros take its positive version. Therefore, we utilize the value $|B| = \frac{1}{4}$ in the period formula:

$\text{Period}=\frac{2\pi}{|B|}$

$\text{Period}=\frac{2\pi}{\frac{1}{4}}$

$\text{Period}=8\pi$

The period of the function is $8\pi$ .

Menses of the sine – Exercise problems

Solve the following practice issues using what yous take learned well-nigh the period of sine functions. If you demand help with this, you lot can look at the solved examples above.