Trigonometric table

In mathematics, tables of trigonometric functions are very helpful in many areas. Before pocket calculators existed, trigonometric tables were extremely important for navigation, science, and engineering. People used these tables to solve problems and make calculations much easier.

The making of these tables was a big job and helped lead to the creation of the first mechanical computing devices. Trigonometric calculations were also key in the early days of studying the stars and planets. These tables were built by using special math rules called trigonometric identities to find new numbers from old ones.

Today, computers and calculators can find these values quickly using special math programs. These programs sometimes still use old tables inside them and find answers using a method called interpolation. This is especially useful in computer graphics, where quick answers are needed and perfect accuracy isn’t always required.

Trigonometric tables are also important for something called the fast Fourier transform (FFT), which is used in many types of data processing. Because the same numbers are needed again and again, scientists have found smart ways to either store these numbers or calculate them quickly as needed. A trigonometry table is like a chart that shows the values of sine, cosine, tangent, and other trigonometric functions for different angles, making it easy to look up the information you need.

Using a trigonometry table

To use a trigonometry table, first decide the angle you need to find values for. Look for this angle along the top row of the table. Next, pick the trigonometric function you need from the first column on the side. Finally, find where your chosen function and angle meet in the table; the number there is the value you're looking for.

On-demand computation

Modern computers and calculators can find trigonometric function values for any angle using different methods. One common way is to use a mathematical shortcut called an approximation along with a small table of angles. They look up the closest angle in the table and then use the approximation to find the exact value.

Another method, called CORDIC, is used on simpler devices. It works by using shifts and additions instead of multiplication, making it faster and easier for the device to calculate.

Example

To calculate the sine of 75 degrees, 9 minutes, and 50 seconds using old trigonometric tables, you would round up to 75 degrees and 10 minutes. You could then look up the value for 10 minutes on the 75-degree page. This method gives an answer accurate to four decimal places.

For more precision, you could use a process called linear interpolation. By comparing the sine values for 75 degrees 10 minutes and 75 degrees 9 minutes from the table, and adjusting for the extra 50 seconds, you can get a more accurate result. This kind of careful calculation was very important for tasks like navigation and astronomy before we had modern calculators.

Half-angle and angle-addition formulas

Historically, trigonometric tables were often made by using special math rules called half-angle and angle-addition formulas. These rules start from a known value, like the sine of a certain angle, and help calculate other values step by step. The ancient astronomer Ptolemy used these methods in his book, the Almagest, to create tables that helped solve problems in astronomy.

These formulas include ways to find the sine and cosine of half an angle or the sum and difference of two angles. They were very important before calculators existed and were used to build early trigonometric tables, like Ptolemy's table of chords. Some tables even used different functions, such as sine and versine, instead of sine and cosine.

A quick, but inaccurate, approximation

A simple way to estimate values for sine and cosine uses a step-by-step process. Start with s₀ = 0 and c₀ = 1. Then, for each step, update the values using special rules involving a small number d. However, this method isn't very precise. For example, when trying to create a table with 256 entries, the biggest mistake in the sine values is about 0.061. With more entries, like 1024, the mistake gets smaller but is still noticeable. This approach would draw a spiral shape instead of a perfect circle if you plotted the points.

The method is related to solving a math problem called a differential equation, which describes how sine and cosine change smoothly over time.

Main article: Euler method
Main articles: Trigonometric functions, Differential equation

A better, but still imperfect, recurrence formula

This section explains a way to create tables of trigonometric numbers using a special math rule. It starts with two simple numbers and uses them to find more values step by step. Even though this method aims for perfect results, small mistakes happen when using computers because they can only handle numbers with limited detail.

A smarter version of this rule helps reduce those mistakes even more, making the results more accurate for big calculations like those used in signal processing. However, some tiny errors can still affect very large calculations.