Beginning Trigonometry - Lesson 1 - Radians and degrees

Introduction

It is important to understand what a radian and degree is, and how they relate to each other. This lesson explains what that are.

Radians and degrees are just different units for measuring an angle,

much like 'cm' and 'inches'. They are directly proportional to each other.

Degrees

Degrees are a unit of measurement that are used to express directionality and the size of an angle. If you stand facing directly north, you are facing the direction of zero degrees, written as 0°. If you turn yourself fully around, so you end up facing north again, you have "turned through" 360°; that is, one revolution (one circle) is 360°.

Below shows a circle split into 360 sections. Each section is 1° (one degree):

circle split into 360 degrees

You may ask yourself how it was decided that a circle should be split into 360 parts - who created this magic number, and why, and why was it called degrees?

Well we can all thank the ancient Babylonians who lived about 5000 years ago. They viewed the numbers 6, 12 and 60 as having particular religious meanings and significance. It is because of them that we have twelve-hour nights and twelve-hour days, with each hour divided into sixty minutes and each minute divided into sixty seconds. And "once around" is divided into 6×60 = 360 parts called "degrees".

So from now one, one thing to remember is when I mention 1 full revolution, I am talking about 360°, and a half revolution would be, you guessed it - 180° - also known as "about face".

90° represents a quarter of a circle.

Understanding initial and terminal angles meaning, and what an arc is

Run the following following scenario in you mind:

1) You are facing north holding your arm out in front of you

2) you turn 90° to face west - sweeping your arm across with you. 

When people talk about the "initial angle" this was your arms starting position (north).

Where your arm ended facing west - that is the "termainal angle"

The action of your arm sweeping from the initial angle to the terminal angle is known to have "swept out" a 90° angle.

If you imaging the line that your fingertips would have draw - the curve - is known as the "arc", and the angle that was turned through is said to "subtended" that arc.

Decimal Degrees and "Degrees, Minutes and Seconds"

When you work with degrees, you'll almost always be working with decimal degrees; that is, with degrees expressed as decimal numbers such as 43.1025°. But just as "1.75" hours can be expressed as "1 hour and 45 minutes", so also "degrees" can be expressed in terms of smaller units. These units, just as for "hours", are called "minutes" and "seconds". Just as "hours" can be expressed as decimals or else as hours - minutes - seconds, so also "degrees" can be expressed as decimals or else as degrees - minutes - seconds, denoted as "DMS".

• Convert 43.1025° to DMS form.

I can see that I have 43°, but what do I do with the "0.1025" part? I treat it like a percentage of the sixty minutes in one degree, and find out how many minutes this is:

(0.1025 degrees)(60 minutes / 1 degree) = 6.15 minutes

...or 6 minutes and 0.15 of a minute. Each minute has sixty seconds, so:

(0.15 minutes)(60 seconds / 1 minute) = 9 seconds

Then 43.1025° = 43° 6' 9"

Notice the symbols: A single quote-mark (an apostrophe) indicates "minutes" and a double quote-mark indicates "seconds". This is similar to the notation (in Imperial measurements) for "feet" and "inches": the smaller unit gets the more-substantial mark.

• Convert 102° 45' 54" to decimal form.

Clearly, I've got 102°, but how do I convert the minutes and seconds to decimal form? By using the definitions and doing the divisions. The 45' means 45/60 of a degree, since each degree contains sixty minutes. Simplification and long division gives me 45/60 = 3/4 = 0.75. So the 45' is 0.75°.
Now I need to deal with the 54". Since each minute is sixty seconds, then I get 54/60 = 9/10 = 0.9. But this is minutes. Now I need to convert the 0.9 of a minute to degrees:

(0.9 minutes)(1 degree / 60 minutes) = 0.015 degrees

So 102° 45' 54" = 102° + 0.75° + 0.015° = 102.765°.

Radians 

Why do we have to learn radians, when we already have perfectly good degrees? Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc).

The 360° for one revolution ("once around") is messy enough. Why is the value for one revolution in radians the irrational value 2π? Because this value makes the math work out right. You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to make sense, the "numerical" value corresponding to 360° needed to be defined as (that is, needed to be invented having the property of) "2π is the numerical value of 'once around'."

Converting Between Radians and Degrees

We know that the circumference of a circle is 2pie r. When we are measuring the angle in radians, we are trying to find the ratio of the arc to the radius.

As a result, the angle subtended at O = 

We also know that the number of degrees in a circle is 360°
Hence, 

or 


It's not necessary to remember what 1 degree or 1 radian is. More importantly, you should understand the concept and derive it on the spot.
Scholar's Tip: When the unit of an angle is not specified, it usually means that the angle is in radians.

Examples

Radians into degrees

a) 2.63 
So we simply multiply 2.63 with  to obtain
157.7°  (correct to the nearest 0.1 degrees) .

Degrees into Radians

b) 37°
We multiply     with 37 to obtain 0.64565   .