There are 360o in
a complete circle

## Units of Measure

We have been measuring angles in degrees, with ${\displaystyle 360^{\circ }}$ in a complete circle.

### Choice of Units for Length and Weight

In measuring many quantities we have a choice of units. For example with distances we can use the metric system and measure in metres, kilometres, centimetres, millimetres. It is also possible to measure distances in miles, yards, feet and inches. With weights we can measure in kilogrammes and grammes. We can also measure in pounds and ounces.

### Choice of Units for Measuring Time

In measuring time we choose to have 60 seconds in a minute and 60 minutes in an hour. We could devise a new more metric system for time and divide an hour into 100 units, each three fifths of our current minute, and then divide these shorter 'minutes' up into 100 units each of which would be about a third of a second.

### Why 60? Why 360?

The choice of dividing into 60 is not entirely arbitrary. 60 can be divided evenly into 2,3,4,5 or 6 or 10 or 12 parts. 60 can't be divided evenly into 7 equal parts, each a whole number in size, but it's still pretty good. Using 360 degrees in a full circle gives us many ways to divide the circle evenly with a whole number of degrees. Nevertheless, we could divide the circle into other numbers of units.

## Choice of Units for Radians

 Exercise: Check the Statements Are the statements: There are about 6.28 Radians to a complete circle. There are about 57.3 degrees in one Radian. Compatible? It is not hard to check. Digression: In maths books it is well worth quickly checking statements that can be checked easily. It helps reinforce your understanding and confirm that you are understanding what is being said. Also, unfortunately, it isn't that unusual for maths books have tiny slips in them, where the person writing the book has written say, ${\displaystyle x_{i}}$ instead of ${\displaystyle x_{j}}$ or some other small slip. These tend to happen where the author knows the material very well and is seeing what he expects to see rather than what is actually written. They can be very confusing to someone new to the material. These kinds of mistakes can also happen in wikibooks, sometimes a visitor trying to improve the content can actually introduce errors. In wikibooks you may also see sudden changes in notation or notation that does not match a diagram, where material has been written by different people.

We said "there are about 6.28 Radians to a complete circle". The exact number is ${\displaystyle 2\pi }$, making the number of radians in a complete circle the same as the length of the circumference of a unit circle.

Remember that:

The circumference of a circle is
${\displaystyle 2\pi \times R}$
where ${\displaystyle R}$ is the radius.

## Justifying Choice of Units for Radians

At this stage in explaining trigonometry it is rather difficult to justify the use of these strange units. There aren't even an exact whole number of radians in a complete circle. In more advanced work, particularly when we use calculus they become the most natural units to use for angles with functions like ${\displaystyle \cos(\alpha )}$ and ${\displaystyle \sin(\alpha )}$ . A flavour of that, but it is only a hint as to why it is a good unit to use, is that for very small angles.

${\displaystyle \sin(\alpha )\approx \alpha }$

And the approximation is better the smaller the angle is. This only works if we choose Radians as our unit of measure and small angles.

 Worked Example: Small angles in Radians and Degrees We claim that for small angles measured in radians the angle measure and the sine of the angle are very similar. Let us take one millionth of a circle. In degrees that is 0.00036 degrees. In Radians that is ${\displaystyle {\frac {2\pi }{1,000,000}}\approx 0.00000628}$ Radians. The angle of course is the same. It's one millionth of a circle, however we choose to measure it. It is just as with weights where a weight is the same whether we measure it in kilogrammes or pounds. The sine of this angle, which is the same value whether we chose to measure the angle in degrees or in radians, it turns out, is about 0.00000628. If your calculator is set to use degrees then ${\displaystyle \sin(0.00036^{\circ })}$ will give you this answer.

There are
${\displaystyle 2\pi }$ Radians
in a complete circle.

It is traditional to measure angles in degrees; there are 360 degrees in a full revolution. In mathematically more advanced work we use a different unit, the radian. This makes no fundamental difference, any more than the laws of physics change if you measure lengths in metres rather than inches. In advanced work, If no unit is given on an angle measure, the angle is assumed to be in radians.

${\displaystyle {\frac {3\pi }{2}}^{c}\equiv {\frac {3\pi }{2}}{\rm {rad.}}\equiv {\frac {3\pi }{2}}}$

A notation used to make it really clear that an angle is being measured in radians is to write 'radians' or just 'rad' after the angle. Very very occasionally you might see a superscript c written above the angle in question.

### What You need to Know

For book one of trigonometry you need to know how to convert from degrees to radians and from radians to degrees. You also need to become familiar with frequently seen angles which you know in terms of degrees, such as ${\displaystyle 90^{\circ }}$ in terms of radians as well (it's ${\displaystyle {\frac {\pi }{2}}}$ Radians). Angles in Radians are nearly always written in terms of multiples of Pi.

You will also need to be familiar with switching your calculator between degrees and radians mode.

Everything that is said about angles in degrees, such as that the angles in a triangle add up to 180 degrees has an equivalent in Radians. The angles in a triangle add up to ${\displaystyle \pi }$ Radians.

A single radian is defined as the angle formed in the minor sector of a circle, where the minor arc length is the same as the radius of the circle.

Defining a radian with respect to the unit circle.

${\displaystyle 1{\rm {rad}}\approx 57.296^{\circ }}$

#### Measuring an angle in radians

The size of an angle, in radians, is the length of the circle arc s divided by the circle radius r.

${\displaystyle {\text{angle in radians}}={\frac {s}{r}}}$

We know the circumference of a circle to be equal to ${\displaystyle 2\pi r}$ , and it follows that a central angle of one full counterclockwise revolution gives an arc length (or circumference) of ${\displaystyle s=2\pi r}$ . Thus ${\displaystyle 2\pi }$ radians corresponds to ${\displaystyle 360^{\circ }}$ , that is, there are ${\displaystyle 2\pi }$ radians in a circle.

## Converting between Radians and Degrees

Because there are ${\displaystyle 2\pi }$ radians in a circle:

${\displaystyle \theta ^{c}=\theta ^{\circ }\times {\frac {\pi }{180}}}$
${\displaystyle \phi ^{\circ }=\phi ^{c}\times {\frac {180}{\pi }}}$
 Conversion from degrees to radians Convert ${\displaystyle 180^{\circ }}$ into radian measure. ${\displaystyle 90^{\circ }}$ into radian measure. ${\displaystyle 90^{\circ }}$ into radian measure. ${\displaystyle 137^{\circ }}$ into radian measure.
 Conversion from radians to degrees Convert: ${\displaystyle {\frac {\pi }{3}}}$ into degree measure. ${\displaystyle {\frac {\pi }{6}}}$ into degree measure. ${\displaystyle {\frac {7\pi }{3}}}$ into degree measure. ${\displaystyle {\frac {3\pi }{4}}}$ into degree measure.