Why trigonometry works

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why do trigonometric functions work?

Ask Question. Asked 6 years ago. Active 5 years, 11 months ago. Viewed times. Sir Cumference Sir Cumference 9 9 bronze badges. That's a Taylor series, and understanding why that series exists and why the coefficients have those values is going to require a lot of calculus.

Show 3 more comments. Active Oldest Votes. Early work with spherical triangles was as important as plane triangles. The first work on trigonometric functions related to chords of a circle. Given a circle of fixed radius, 60 units were often used in early calculations, then the problem was to find the length of the chord subtended by a given angle. The first known table of chords was produced by the Greek mathematician Hipparchus in about BC.

Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry. You can also work out the inverse function to sin, cos and tan, which means 1 divided by that function.

This enables you to work out the angle if you have the sin, cos or tan of it. Scientific calculators have sin, cos, and tan functions, as well as inverse functions. Trigonometry also works for other triangles, just not in quite the same way. Instead there are two rules based on a triangle like this:. This is a reasonable question, and the answer is at least partly because those who decide the mathematics curriculum in many countries think that you should know about it, and for very good reason.

Trigonometry is said to be the most important mathematical relationship ever discovered. Triangles are one of the most simple forms found in nature, but their mathematics has vital importance, especially where precise distance measurements are needed.

When we begin to think about the applications where accurate distances are important, it is apparent that there are dozens, including navigation in naval and aviation systems, astronomy, satellite systems, geographical surveys and cartography maps , architecture and structural engineering, graphic design and computer generated imagery. Many of these rely on a measurement technique known as triangulation , which applies the concepts of trigonometry.

You can be motoring in one direction, but the tide could be coming from one side, and push you to the other. You will need trigonometry to work out how far you will travel and in what precise direction. What direction will you end up travelling in? First draw your triangle , and label the sides.

You have the opposite and the adjacent, which means that you need to use tangent. Now is the time to use the inverse tan function. The inverse tan of 0. In other words, tan Why is this important? Trigonometry may not have all that many everyday applications, but it does help you to work with triangles more readily. Continue to: Geometry Introduction to Algebra.