What’s my Angle?

This is the first post in the trigonometry category. Trigonometry is the study of the properties of triangles: the lengths of their sides and the relationships with their angles. So the first topic in trigonometry is to define what an angle is.

An angle is a measure of rotation of one line from another where these lines are connected at one end. Like x is used to commonly refer to an unknown or general number in algebra, 𝜃, which is the greek letter “theta”, is commonly used to represent a generic angle. I presume this is so because the ancient Greeks did a lot of work in the fields of trigonometry and geometry. Below shows a generic angle between two lines:

Now there are several units used to measure angles. The one most known to most non-maths people is degrees. There are other measures: radians which is used frequently in maths, and gradians which is sometimes used in engineering and surveying. Here, we will use degrees.

The point where the lines join is called the vertex. You can imagine it as a flexible joint where the top line pivots around the horizontal line from being right on top of the horizontal line to swinging all the way around so that it again is superimposed on the horizontal line. When the lines are first superimposed, this is a 0 degree angle. The notation for degrees is a small circle at the upper right of the number: 0°. When the line does a full rotation so that it is again superimposed, that angle is 360°.

Now you can continue rotating the line and think of angles greater than 360°, but this post will limit itself to angles between 0° and 360°. I am specifically interested in the angle when the rotating line is pointing straight up, that is becomes perfectly vertical. Well this is a quarter of the way around to 360° so a quarter of 360 is 90. The angle is now:

Note that it is common to add lines in this angle to form a small square when the angle is 90°. This angle holds special interest in maths and is given a name: right angle. The term “right” comes from latin meaning “upright”.

This angle will be used in my next post on triangles.