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# Sine Wave: Definition, What It's Used For, Example, and Causes

## What Is a Sine Wave?

A sine wave is a geometric waveform that oscillates (moves up, down, or side-to-side) periodically, and is defined by the function y = sin x. In other words, it is an s-shaped, smooth wave that oscillates above and below zero.

Sine waves are used in technical analysis and trading to help identify patterns and cross-overs related to oscillators.

### Key Takeaways

• A sine wave is an S-shaped waveform defined by the mathematical function y = sin x.
• It is depicted graphically as two semi-circular curves that alternate above and below a center line.
• In finance, market participants may identify cyclical patterns or oscillator signals from sine-wave-based functions.
• The sine wave as a technical chart analysis tool is based on advanced mathematics and is designed to indicate whether a market is trending or in a cycle mode.
• Sine waves are the basis of Fourier decomposition analysis.

## Understanding Sine Waves

The sine wave indicator is based on the assumption that markets move in cyclical patterns. After quantifying a cycle, a trader may try to use the pattern to develop a leading indicator. This works extremely well when the market is indeed moving in a cycle. When the market is trending, however, this system fails (and one should adjust for that).

Markets alternate between periods of cycling and trending. Cyclical periods are characterized by price bouncing off support or resistance levels and failed breakouts or overshoots. Trending periods are characterized by new highs or new lows and pullbacks that then continue in the direction of the trend, until exhausted.

In technical analysis, oscillators are often used that may have sine-shaped characteristics. An oscillator exists between two extreme values and then builds a trend indicator with the results. The analysts then use the trend indicator to discover short-term overbought or oversold conditions. When the value of the oscillator approaches the upper extreme value, analysts interpret that information to mean that the asset is overbought, and as it approaches the lower extreme, analysts consider the asset to be oversold.

The complement to a sine function is the cosine.

## Sine Waves as Analytical Tools

The sine wave as a technical chart analysis tool is based on advanced mathematics and is designed to indicate whether a market is trending or in a cycle mode. It helps traders identify the start and finish of a trending move as well as possible shifts in the trend.

This leading indicator is also called the MESA indicator and was developed by John Ehlers based on an algorithm that was originally applied to digital signal processing. It consists of two lines, called the Sine Wave and the Lead Wave. When the price is trending, the lines do not cross and usually run parallel and distant from each other.

Line crossovers could indicate turning points and generate buy or sell signals under the right conditions. The indicator can also signal an overbought or oversold market (i.e., unjustifiably high or unjustifiably low), which can have implications on the prevailing trend. Whether used alone or in combination with other techniques or non-correlated indicators (such as moving average-based indicators), the sine waves are very useful for a trader.

The Composite Index of Lagging Indicators often resembles a sine wave since the measures that make up the index (i.e., ratios and interest rates) tend to oscillate between a range of values.

For example, inflation is always kept between specified rates and if/once inflation meets or exceeds a specified limit, interest rates will be adjusted to either increase or decrease inflation so it is brought within a target range. Thus, as the rate of inflation increases, decreases, or stays the same, interest rates will oscillate up and down to control an undesired rate of inflation.

## Who Uses Sine Waves in the Markets?

Technical traders who employ Fourier analysis are inherently using sine waves to guide their trading decisions. Fourier analysis is a technique that essentially breaks down a piece of complex time series data into a series of simpler components that are based on the sine or other trigonometric functions. By doing, a trader is better able to separate the signal from the noise in the data.

For instance, if a trader believes that inflation is cyclical and presents noise in the price history of a particular consumer cyclical stock, they can try to identify a sine wave that fits the inflation pattern and remove it. While Fourier analysis has been applied to by market technicians, financial researchers remain unconvinced that it is a viable or effective strategy.

## How Can a Sine Curve Describe a Wave?

A wave (whether a sound wave, ocean wave, radio wave, or any other) can be described by its amplitude (height or power) and frequency (how close together each wave peak is from the next). In doing so, a sine curve is generated of a particular height and frequency.

## How Do I Graph Sine and Cosine Functions?

A cosine curve is graphed similarly to a sine curve, but is out of phase with it. In particular, a sine wave crosses the center line at each interval of pi (π), whereas a cosine wave peaks at intervals of π and crosses the center line at intervals of ½π (which are the points where sine waves peak).

You can graph specific sine or cosine curves using a graphing calculator, mathematical or spreadsheet software like Excel, or via one of several online tools.

## How Do I Find the Period of a Sine Wave?

The period of a sine curve is the length of a single wave from the center like to peak, through the center to trough, back to center. It is expressed mathematically as period = 2π/|B|, where B is the horizontal stretch of each wave in the sine function.

Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
1. ResearchGate. "Fourier Analysis for Stock Price Forecasting: Assumption and Evidence." Accessed Jan. 4, 2022.

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