怎样使用三角函数

Wikipedia says:

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Sal Khan says:

Trigonometry is about ratios of sides of triangles.

Really, they are almost the same thing and you will learn, in trigonometry, how angles, side lengths, triangles, and circles are all related and you can get one from the other. Trigonometry combines algebraic and trigonometric properties and identities and makes a whole new subject that is worth exploring. Move ahead in the map to learn more!

In this post, we will first review slopes, and then mention an interpretation using trig.

Review of slopes

Given two points on a line $(x_1, y_1) (x_2, y_2)$ we can find the slope of the line is $\frac{y_2-y_1}{x_2-x_1}$.

This number basically tells us how fast the line is increasing.

If a line is in the form $y=mx+b$, then we can find $m$ is the slope of the line.

Trig interpretation of slope

Slope can also be defined in a different way: If we have a line going through the origin, and it has argument $\theta$, then the slope is equal to $\tan \theta$.

For example, if a line is at a 45 degree angle, then the slope is obviously $\tan 45 = 1$.

In general, trigonometry involving right triangles is the most fundamental of all areas and applications of trigonometry, and serves as a basis for other applications.

Since all polygons can be decomposed into trangles, and all triangles can be further decomposed into right triangles, trigonometry involving right triangles can be applied to almost all polygons.

For example, take a regular hexagon. It can be evenly divided into six isosceles triangles, which can in turn be made into 12 right triangles. So trigonometry directly applies to some regular hexagons since they are made of 12 right triangles.

To be updated and expanded.

The word trigonometry has greek roots which are trigon, which means triangle, and metry, which means measure. So just from the word trigonometry one would say that the subject is about triangles and measuring them. This is absolutely correct!

Trigonometry was first used by the ancients to pull off extraordinary feats! It was used to build the great pyramids of Egypt and pretty much every remarkable engineering feat in the ancient world!

As you will learn trigonometry is not just restrained to the realm of triangles, but gives birth to new functions and really elegant and beautiful theorems and identities. Perhaps you have heard of Euler's famous identity which is $e^{i\pi}=-1$.

Hopefully you are now as excited about trigonometry as I am and will continue to learn this truly beautiful subject!

Trig is often useful when you have perpendiculars (which give you right triangles to apply right triangle trig to).

Sometimes these perpendiculars are given to you (Example: Angles of elevation and depression). Sometimes, you can create these perpendiculars yourself (like for the law of cosines).