## Radians

Radians are one way of calculating angles.

There are 4 common ways:

- Radians

- Degrees

- Gradians (basically, 360 degrees = 400 gradians)

- turns - 360 degrees = a turn

From this, we can see that the most complicated type, is in fact, radians.

A radian is the ratio between the length of an arc and it's radius.

i.e. ----------> length of subtended (by radius) arc / radius ---------------> s / r in the diagram below)----------------------> gives us radians

Hence, you get 1 radian or "1 rad" when length of subtended arc = radius

There are 4 common ways:

- Radians

- Degrees

- Gradians (basically, 360 degrees = 400 gradians)

- turns - 360 degrees = a turn

From this, we can see that the most complicated type, is in fact, radians.

A radian is the ratio between the length of an arc and it's radius.

i.e. ----------> length of subtended (by radius) arc / radius ---------------> s / r in the diagram below)----------------------> gives us radians

Hence, you get 1 radian or "1 rad" when length of subtended arc = radius

From the picture,

We see that s is the subtended arc and AO and BO subtend it. That's what subtend means... to like erm... "mark the boundary of"

When s = r, s/r = r/r = 1 radian

Now, many people say that 2pi = 360 degrees in trigonometry. But why is this so?

This is explained by the derivation or radians.

We see that s is the subtended arc and AO and BO subtend it. That's what subtend means... to like erm... "mark the boundary of"

When s = r, s/r = r/r = 1 radian

Now, many people say that 2pi = 360 degrees in trigonometry. But why is this so?

This is explained by the derivation or radians.

Take a look at the diagram again. Notice that the angle controls the size of s, the subtended arc.

When tita, the angle is bigger, s is also bigger. When tita is smaller, s will also be smaller. The angle and subtended arc are thus proportional.

Think about it this way,

when tita is 90 degrees, AOB will be a quarter-circle --- making s a quarter of the total circumference.

Let's jump a bit here.

When tita, the angle is bigger, s is also bigger. When tita is smaller, s will also be smaller. The angle and subtended arc are thus proportional.

Think about it this way,

when tita is 90 degrees, AOB will be a quarter-circle --- making s a quarter of the total circumference.

Let's jump a bit here.

By definition,

Since 2(pi)r = circumference of a circle, where according to the diagram, tita = 360. Here, we see that y = 2(pi)r

2(pi)r / r radians = 2(pi) radians

Therefore, 2(pi)radians = 360 degrees.

Dividing 2(pi) on both sides gives

1 radian = 360/2(pi) = 180/pi

Hence,180 = pi, 360 = 2(pi)

Since 2(pi)r = circumference of a circle, where according to the diagram, tita = 360. Here, we see that y = 2(pi)r

2(pi)r / r radians = 2(pi) radians

Therefore, 2(pi)radians = 360 degrees.

Dividing 2(pi) on both sides gives

1 radian = 360/2(pi) = 180/pi

Hence,180 = pi, 360 = 2(pi)