Period Of Cos X 2
The cosine function is a trigonometric role that is periodic. A periodic office is a function that repeats itself over and over in both directions. The catamenia of the cosine function is 2π, therefore, the value of the function is equivalent every 2π units. For example, we know that we have cos(π) = 1.
Every time we add 2π to the10 values of the function, we accept cos(π+2π). This is equivalent to cos(3π). We have the consequence cos(π)=1 and since the function is periodic, we also have the event cos(3π) = i.
TRIGONOMETRY
Relevant for…
Learning to find the flow of the cosine function.
See examples
TRIGONOMETRY
Relevant for…
Learning to observe the menses of the cosine function.
Meet examples
Period of the basic cosine role
The cosine function in its well-nigh basic grade is $latex y=\cos(10)$. This part can be evaluated for whatever real value, so we tin can use all existent values ofx. This ways that the role extends indefinitely to the right and to the left.
Using a graph of the cosine function, we can make up one's mind its period by looking at the distance between "equivalent" points. That is, the menses of the role $latex y = \cos(ten)$ is the distance on thex-axis between repeating patterns.
We can hands see that the graph repeats subsequently 2π. Therefore, nosotros conclude that the period of the role is 2π. The reason nosotros have this flow is that in the unit circle, 2π equals 1 complete revolution around the circumvolve.
This ways that if we take a value greater than 2π, nosotros would simply be repeating the loop around the unit circle and we would obtain values equivalent to the angles between 0 and 2π.
Period of other variations of the cosine function
The catamenia of the cosine role in its basic course, $latex y = \cos(ten)$, is 2π. This period can exist modified past multiplying the variablex by a constant.
We can reduce the menses of the part past multiplyingten by a number greater than ane. This will cause the role to exist "sped up" and the flow to go smaller. This means that the function volition occur more quickly and information technology volition take less for it to start repeating itself.
For case, in the function $latex y = \cos (2x)$, the period is π, which is half the menses of the original function.
When we multiply the variable10 by a fractional number that is greater than 0 and less than i, we will make the function reduce its "speed" and brand it have a larger menses. This means that the function volition take longer to first repeating itself.
For example, in the function $latex y = \cos(\frac{10}{two})$, the catamenia is 4π, which is twice the menstruation of the original office.
How to determine the period of a cosine function?
We can decide the menstruation of a cosine part by using the coefficient of the variable10. This coefficient is usually represented by the alphabetic characterB. Therefore, the standard grade of the cosine function is $latex y = \sin(Bx)$. Using this form, we can obtain the following formula:
$latex \text{Menses}=\frac{ii\pi}{|B|}$
This ways that to obtain the period, nosotros only accept to divide 2π by |B|, where, |B| is the absolute value ofB. To find the absolute value, we simply take to take the positive version of the number. For instance, if we have -2, its absolute value is two.
We tin can utilise this formula fifty-fifty when nosotros accept other variations of the cosine function. For example, if we accept the office $latex y = 2 \cos(2x+5)$, we but take the coefficient of the variablex:
$latex \text{Period}=\frac{two\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{2}$
$latex \text{Catamenia}=\pi$
Period of the cosine role – Examples with answers
The following examples are solved using the formula for the flow of cosine functions. Each example has its corresponding solution, but it is recommended that y'all try to solve the bug yourself before looking at the solution.
Example one
- If we have the function $latex y = \cos(3x)$, what is its period?
Solution: We tin can place the value $latex |B|=5$. Using this value in the formula for the menses, we have:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{two\pi}{v}$
The period of the role is $latex \frac{2}{5}\pi$.
EXAMPLE ii
- What is the period of the cosine function $latex y=2 \cos(4x)-iii$?
Solution: The part has a more complex form than the previous i, simply we only need the coefficient ofx. Therefore, we recognize the value $latex |B|=iv$ and use information technology in the period formula:
$latex \text{Flow}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{4}$
$latex \text{Period}=\frac{\pi}{2}$
The menses of this function is $latex \frac{\pi}{2}$.
EXAMPLE 3
- What is the period of the part $latex y = \frac{1}{three}(- \frac{1}{5} x-two)$?
Solution: Again, we just use the coefficient ofx to find the period. In this case, the coefficient is negative, and so we only accept its positive value. Therefore, we use the value $latex |B| = \frac{1}{5}$ in the menstruum formula:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Menstruum}=\frac{2\pi}{\frac{i}{5}}$
$latex \text{Menstruum}=10\pi$
The period of the role is $latex 10\pi$.
Period of the cosine – Exercise bug
Use what you lot accept learned to solve the following exercises for menstruum of cosine functions. If y'all need help with this, you can await at the solved examples above.
What is the period of the function $latex y=\cos(3x)$?
Cull an answer
If we take the role $latex y=two\cos(\frac{two}{3}ten-2)$, what is its period?
Cull an answer
What office has a period of $latex 7\pi$?
Cull an answer
Run into also
Interested in learning more than about the cosine of an bending? Accept a look at these pages:
- Cosine of an Angle – Formulas and Examples
- Graph of Cosine with Examples
- Amplitude of Cosine Functions – Formulas and Examples
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Period Of Cos X 2,
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