The Phase Shift Calculator offers a quick and free solution for calculating the phase shift of trigonometric functions. In order to comprehend better the matter discussed in this article, we recommend checking out these calculators first Trigonometry Calculator and Trigonometric Functions Calculator.

Trigonometry is encharged in finding an angle, measured in degrees or radians, and missing sides of a triangle – basic functions in trigonometry are sine, cosine, and tangent. Also, suppose you seek to enlarge your knowledge. In that case, even more, you shouldn’t miss using these calculators too: Law of Sines Calculator, Law of Cosine Calculator, Cosine Calculator, Sine Calculator, and Cofunction Calculator – sin, cos, tan, cot, sec, csc. Contact us if there is anything we can help you with your studying!

Besides calculating the phase shift of trigonometric functions, we will also cover the amplitude, frequency, and period. By saying so, we mean that we will provide a detailed explanation of these terms and all others closely related to this topic, followed by an example and step-by-step tutorial on how to find each. However, if your only aspiration is to learn how to handle our calculator, scroll to the end of the page and check it out.

Amplitude, Period, and Phase Shift

To make it easier for you to understand the definitions that will be provided for these terms and, also, the difference among them, we will start with the graph that includes them.

Phase shift calculator
Phase shift, period, amplitude, and vertical shift

The amplitude of a function is the distance from the highest point of the curve to the midline of the graph. While the midline is a horizontal axis that serves as the reference line around whom the curve of a periodic function oscillates.

Also, the midline is placed where the average of the minimum and maximum values of the function is. In other words, the amplitude is half of the distance measured between the upper highest point of the curve and the bottom highest point on a graph.

The period is defined as the length of the function’s cycle, which means that the distance between the repetition of any function is called its period. Basic trigonometric functions such as sine, cosine, secant, and cosecant have a period of 2\pi , while tangent and cotangent have a period of \pi .

Frequency is the term used to show how often the graph oscillates around the axis, measured per unit of time. There is a difference between the period and frequency, but they are related: Frequency= \frac{1}{Period} .

The phase shift means that the function’s graph is shifted or displaced left or right from its usual position. It also implies how far the function is shifted horizontally from the original position (delay). In most cases, functions are shifted \frac{\pi}{2} from the usual position. In other words, this phenomenon indicates that two signals at a given time are at different points of their cycle.

Therefore, the phase shift is measured as the angle, whether in degrees or radians, between two points on a circle simultaneously while demonstrating the progress of each wave through its cycle.

The Phase Shift Formula

The phase shift equation:

f(x)= A \cdot sin(Bx - C) + D
f(x)=A \cdot cos(Bx - C) + D

Where A, B, C, and D are arbitrary real numbers, where each represents:

  • A – amplitude,
  • B – period,
  • D – vertical shift, and
  • \frac{C}{B} – phase shift – measured as an angle, in degrees or radians.

If \frac{C}{B} is positive, the curve moves right and negative, the curve moves left.

How to Calculate the Phase Shift?

The phase shift is the horizontal translation of the function concerning the regular sin(x) or cos(x), measured as an angle whose phase shift is equal to 0. By comparing the graphs of their functions, we couldn’t but notice that we could get one by translating the other. Spoken in trigonometry terms: 

sin(x+ \frac{\pi}{2})=cos(x)
cos(x- \frac{\pi}{2})=sin(x)

Let’s find the letters responsible for the phase shift.

The phase shift formula gives:

A \cdot sin(Bx-C)+D=A \cdot sin(B \cdot (x- \frac{C}{B}))+D ,

which is a phase shift of ( \frac{C}{B} ) of the function A \cdot sin(Bx) .

To sum it all up, to find the phase shift in a phase shift formula, you will need to calculate ( \frac{C}{B} ).

How to Find the Amplitude?

The sine and cosine functions have amplitude values ranging from [-1 to 1]. That does not affect if we replace:

  • sin(x) or cos(x) for sin(Bx-C) , or
  • cos(Bx-C) for a B\neq 0 and arbitrary C.

 Actually, that is because f(x)= Bx-C is then a bijective function. And, if we add D, we have sin(Bx-C)+D or cos(Bx-C)+D . Since the sin(Bx-C) or cos(Bx-C) ranges from [-1 to 1], the equation will be between (-1+D) and (1+D) .

The text from the previous paragraph implies that the midline falls at the same point as D, while the amplitude still equals one because the values are distanced from D as far as one. Accordingly, in the phase shift formulas A \cdot sin(Bx-C)+D and A \cdot cos(Bx-C)+D , the amplitude is affected only if the multiplier (A\neq1), which changes the range to (-1 \cdot A=-A) and (1 \cdot A=A). To summarize, the amplitude in the phase shift formula equals A.

How to Find the Period?

As previously stated, the sine and cosine functions have periods equal to 2π. Therefore, for any x, we have:

  • sin(x+2π)=sin(x) and
  • cos(x+2π)=cos(x) .

So, that results with:

  • A \cdot sin(x+2π)+D=A \cdot sin(x)+D , and
  • A \cdot cos(x+2π)+D=A \cdot cos(x)+D .

As we can see, the A and D in this formula do not affect the period. This leaves us thinking that the conclusion lies inside the trigonometric functions.

But still, sin(x-C+2π)=sin(x-C) and cos(x-C+2π)=cos(x-C) shows that C is not the one for the job. Whatsoever, it does not take someone to be Einstein to figure out that our last option is the one we looked for (B). Now, let’s unwrap the mystery to see how and why B affects periodicity in the phase shift formulas:

sin(Bx)=sin(Bx+2\pi)=sin(B \cdot (x+ \frac{2\pi}{B}))

By each \frac{2\pi}{B} added to the argument x , we are back in the same spot while the function repeats itself. Soaking all of this in, we can acknowledge that the period equals \frac{2\pi}{B} .

How to Find the Vertical Shift?

Let’s start this paragraph by bringing out some often used facts:

  • The phase shift does not affect the vertical shift since they are two perpendicular directions.
  • The amplitude shows us how far the curve can reach vertically, but it can’t shift it.

This leaves us with the letter D, denoting the vertical shift. And the vertical shift is responsible for determining the function’s range.

Phase Shift Calculator – How to Use?

This calculator operates in a manner equivalent to what we described previously that is done by hand. Naturally, a safer way is to use our calculator to avoid mathematical mistakes or forgetting the formula, while your only job is to enter values. So, it only remains for you to enjoy our easy and concise instructions on how to use the phase shift calculator:

  1. After opening our calculator, underneath the picture that describes the parameters discussed in this topic, you have to choose the function that appears in your equation (sine or cosine).
  2. Now, your job is to enter the data you have. But, even without the data provided, our calculator shows the graph for sin(x) or cos(x), depending on the function you have chosen.
  3. Within milliseconds after inputting the values, the function’s graph will appear together with the amplitude, period, phase shift, and vertical shift.
  4. There is one more thing worth attention. Namely, there is an advanced mode allowing you to find the value of a function at any point x_{o} .

Phase Shift Calculator – Example

Find the phase shift of f(x)=3sin(2x+4).

In the formula,

f(x)=Asin(Bx−C)+D,
\frac{C}{B}= \frac{-4}{2}=-2

represents the phase shift.

Using the Phase Shift Calculator saves you time and energy from doing exhausting calculations, which is unnecessary because you have this free tool that provides you with a result of the wanted value within seconds.

FAQ

What is the Phase Shift of a Function?

The Phase Shift represents the horizontal displacement of the function from the usual position measured as an angle. In contrast, the Vertical Shift shows how far the function is shifted vertically from the usual position.

How do you find amplitude and period?

Amplitude measures the distance between the midline of the function and the highest or lowest points of the function. The period can be described as the distance between two graph peaks. In other words, it is the distance between the repetition of that function. Further instructions are given in the text above.

Is phase shift always positive?

If B>0, the phase shift is negative; if B<0, the phase shift is positive (B – period).

How do I find the phase shift from a graph?

First, establish whether it is shifted sine or cosine. Afterward, look at the right side of the graph and try to find the first trough (the coefficient before the function is negative) or peak (the coefficient is positive). Now, calculate the from that point to the vertical axis, and if it was the sine function, subtract \frac {\pi}{2} from it.

What is the phase shift of a periodic function?

The number of cycles of the function that occur in one horizontal unit represents the phase shift of a periodic function.