Find The Period And Domain_3

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

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

TRIGONOMETRY

Relevant for…

Learning to find the period of the sine function.

See examples

TRIGONOMETRY

Relevant for…

Learning to find the period of the sine office.

See examples

Catamenia of the bones sine function

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

Since the graph of $latex y = \sin(x)$ looks like a single blueprint that repeats itself over and over, we can think of the period as the distance on the x-axis earlier the graph starts repeating.

diagram of the period of the sine function

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

Whatsoever quantity greater than 2π ways that nosotros are repeating the revolution. This is the reason why the value of the function is the same every 2π.

Changing the period of the sine part

The period of the bones sine function $latex y = \sin (x)$ is 2π, but if10 is multiplied past a constant, the period of the function tin change.

Iften is multiplied past a number greater than 1, that "speeds upward" the role and the menstruum will be smaller. That means it won't take long for the role to get-go repeating itself.

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

On the other manus, ifx is multiplied by a number between 0 and i, this causes the function to slow downwards and the period will exist larger since information technology will have longer for the function to repeat itself. For instance, the office $latex y = \sin (\frac{x}{2})$ halves the "speed" of the original office. The period of this function is 4π.

Finding the menstruation of a sine office

To observe the period of a sine function, we have to consider the coefficient ofx that is inside the function. We can use B to represent this coefficient. Therefore, if we have an equation in the form $latex y = \sin(Bx)$, we take the following formula:

$latex \text{Menses}=\frac{two\pi}{|B|}$

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

This formula works even if we have more than complex variations of the sine function like $latex y = iii \sin(2x + 4)$. Just the coefficient ofx matters when calculating the flow, and so nosotros would take:

$latex \text{Menses}=\frac{2\pi}{|B|}$

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

$latex \text{Period}=\pi$

Period of the sine role – Examples with answers

What y'all have learned about the catamenia of sine functions is used to solve the following examples. Try to solve the issues yourself before looking at the reply.

Case 1

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

Solution

Nosotros use the period formula with the value $latex | B | = 3$. Therefore, we have:

$latex \text{Menstruation}=\frac{ii\pi}{|B|}$

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

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

EXAMPLE 2

Nosotros take the sine function $latex y = three \sin(4x)+one$. What is its period?

Solution

The only value we demand is the coefficient often. Therefore, we employ the value $latex | B | = 4$ in the formula for the menses:

$latex \text{Catamenia}=\frac{2\pi}{|B|}$

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

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

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

Example three

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

Solution

Nosotros just accept to use the coefficient ofx to find the period. We run into that in this case, the coefficient is negative, then we take its positive version. Therefore, nosotros use the value $latex |B| = \frac{1}{iv}$ in the flow formula:

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

$latex \text{Catamenia}=\frac{2\pi}{\frac{one}{4}}$

$latex \text{Period}=viii\pi$

The period of the function is $latex viii\pi$.

Flow of the sine – Practise bug

Solve the following practise bug using what y'all have learned about the period of sine functions. If y'all need assist with this, yous can look at the solved examples above.

If we take the function $latex y=\sin(5x)$, what is its period?

Choose an answer

$latex \frac{5\pi}{2}$

$latex five\pi$

$latex \frac{2\pi}{5}$

$latex \frac{\pi}{5}$

What is the menses of the function $latex y=four\sin(\frac{2}{3}x)$?

Choose an answer

$latex \frac{iii\pi}{2}$

$latex \frac{\pi}{three}$

$latex \frac{2\pi}{3}$

$latex 3\pi$

Which of the post-obit functions has a period of $latex 5\pi$?

Cull an reply

$latex y=\sin(5x)$

$latex y=5\sin(\frac{1}{five}x)$

$latex y=3\sin(\frac{2}{five}x)$

$latex y=2\sin(\frac{v}{2}x)$

See besides

Interested in learning more well-nigh sine of an angle? Take a expect at these pages:

Sine of an Bending – Formulas and Examples
Graph of Sine with Examples
Amplitude of Sine Functions – Formulas and Examples

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