# Measuring Angles

When two lines intersect, they give rise to four different spaces with respect to the point of intersection.

The spaces created are called **angles**.

The figure shows 4 different angles formed at the intersection of the two lines AB and CD

Angles are commonly measured in degrees denoted as °.
When an object moves through a complete cycle, that is from point D through B, C, A then back to D, we say it has covered 360 degrees (360°).
Therefore a degree is $\frac{1}{360}$ of a cycle.

## Angles greater than 360 degree

We have discussed that when an object makes one complete cycle around a point, it covers 360°, however, when an object makes more than one cycle, it makes an angle greater than 360 degrees. This is a common occurrence in day to day life. A tire makes numerous cycles when a vehicle is being driven hence; it makes an angle greater than 360°.

To find out the number of cycles made when an object rotates, we count the number of times 360 can be added to it to get a number equal to or less than the angle given. Likewise, we find the number than can be multiplied by 360 to get a number less than but closer to the angle given.

**Example 2**

1. Find out the number of cycles made when an object makes an angle of

a) 380°

b) 770°

c) 1000°

__Solution__

a) 380 = (1 × 360) + 20

The object makes one cycle and 20°

Since $20^{\circ} = \frac{20}{360} = \frac{1}{18}$ cycles

The object makes $1\frac{1}{18}$ cycles.

b) 2 × 360 = 720

770 = (2 × 360) + 50

The object makes two cycles and 50°

$50^{\circ} = \frac{50}{360} = \frac{5}{36}$ cycles

The object makes $2\frac{5}{36}$ cycles

c)2 × 360 = 720

1000 = (2 × 360) + 280

$280^{\circ} = \frac{260}{360} = \frac{7}{9}$ cycles

The object makes $2\frac{7}{9}$ cycles

## Positive and negative angles

The figure below shows the sign of an angle measured from a common line, 0 degree line

When an object rotates in an clockwise direction, it makes a **negative** angle of rotation while when it rotates in anticlockwise direction, it makes a **positive** angle.
So far, in our discussions, we have been looking at positive angles only.

In diagram form, a negative angle can be as shown below.

This implies that given a negative angle, we can get its respective positive angle.

For instance, the bottom part of the vertical line is 270°. When measured in the negative direction its will be -90°. We simply subtract 270 from 360.
Given a negative angle, we add 360 to get its corresponding positive angle.

When an angle is -360°, it implies that the object made more than one cycle in clockwise direction.

**Example 3**

1. Find the corresponding positive angle of

a) -35°

b) -60°

c) -180°

d) - 670°

2. Find the corresponding negative angle of 80°, 167°, 330°and 1300°.

__Solution__

1. We add 360 to the angle to get its corresponding positive angle.

a) -35°= 360 + (-35) = 360 - 35 = 325°

b) -60°= 360 + (-60) = 360 - 60 = 300°

c) -180°= 360 + (-180) = 360 - 180 = 180°

d) -670°= 360 + (-670) = -310

That is one cycle in clockwise direction (360)

360 + (-310) = 50°

The angle is 360 + 50 = 410°

2. We subtract 360 from the angle to get its corresponding negative angle.

80° = 80 - 360 = - 280°

167° = 167 - 360 = -193°

330° = 330 - 360 = -30°

1300° = 1300 - 360 = 940 (one cycle made)

940 - 360 = 580 (second cycle made)

580 - 360 = 220 (third cycle made)

220 - 360 = -140°

The angle is -360 - 360 - 360 - 140 = -1220°

Thus 1300° = -1220°

## Radian

A radian is an angle made at the center of circle by an arc which is equal to the length of the radius of that particular circle.
It is therefore a unit that is used to measure an angle. The angle is approximately to 57.3°.

In most cases, it is denoted by **rad**.

Thus $1 rad \approx 57.3^{\circ}$

Radius = r = OA = OB = AB

Angle BOA is equal to 1 radians

Since a circumference is given by $2\pi r$ or $2\pi$ radius, hence there are $2\pi$ radians in one complete cycle.

Radians are commonly given in terms of $\pi$ to avoid dealing with decimals in calculations. In most books, the abbreviation *rad*
is not provided, but the reader has to know that when talking about an angle that is given in terms of $\pi$,
the units are automatically radians.

Some basic angles in radians:

$360^{\circ} = 2\pi\ rad$

$180^{\circ} = \pi\ rad$

$90^{\circ} = \frac{\pi}{2} rad$

$30^{\circ} = \frac{30}{180}\pi = \frac{\pi}{6} rad$

$45^{\circ} = \frac{45}{180}\pi = \frac{\pi}{4} rad$

$60^{\circ} = \frac{60}{180}\pi = \frac{\pi}{3} rad$

$270^{\circ} = \frac{270}{180}\pi = \frac{27}{18}\pi = 1\frac{1}{2}\pi\ rad$

**Example 4**

1. Convert 240°, 45°, 270°, 750° and 390° into radians in terms of $\pi$.

__Solution__

We multiply the angles by $\frac{\pi}{180}$.

$240^{\circ} = 240 \times \frac{\pi}{180} = \frac{4}{3}\pi=1\frac{1}{3}\pi$

$120^{\circ} = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$

$270^{\circ} = 270 \times \frac{1}{180}\pi = \frac{3}{2}\pi=1\frac{1}{2}\pi$

$750^{\circ} = 750 \times \frac{1}{180}\pi = \frac{25}{6}\pi=4\frac{1}{6}\pi$

$390^{\circ} = 390 \times \frac{1}{180}\pi = \frac{13}{6}\pi=2\frac{1}{6}\pi$

2. Convert the following angles into degrees.

a) $\frac{5}{4}\pi$

b) $3.12\pi$

c) 2.4 radians

__Solution__

$180^{\circ} = \pi$

a) $\frac{5}{4} \pi = \frac{5}{4} \times 180 = 225^{\circ}$

b) $3.12\pi = 3.12 \times 180 = 561.6^{\circ}$

c) 1 rad = 57.3°

$2.4 = \frac{2.4 \times 57.3}{1} = 137.52$

### Negative angles and angles greater than $2\pi$ radians

To convert a negative angle to a positive, we add $2\pi$ to the it.

To convert a positive angle to a negative, we subtract $2\pi$ from the it.

**Example 5**

1. Convert $-\frac{3}{4}\pi$ and $-\frac{5}{7}\pi$ to positive angles in radians.

__Solution__

We add $2\pi$ to the angle

$-\frac{3}{4}\pi = -\frac{3}{4}\pi + 2\pi = \frac{5}{4}\pi = 1\frac{1}{4}\pi$

$-\frac{5}{7}\pi = -\frac{5}{7}\pi + 2\pi = \frac{9}{7}\pi = 1\frac{2}{7}\pi$

When an object rotates through an angle that is greater than $2\pi$; it would have made more than one cycle.

To determine the number of cycles of such angle, we find a number when multiplied by $2\pi$, the result is equal or less but closer to the number.

**Example 6**

1. Find the number of cycles made when an objects rotates through the following angles

a) $-10\pi$

b) $9\pi$

c) $\frac{7}{2}\pi$

__Solution__

a) $-10\pi = 5(-2\pi)$; since $-2\pi$ implies one cycle in clockwise direction, it implies that

the object made 5 cycles in the clockwise direction.

b) $9\pi = 4(2\pi) + \pi$, $\pi =$ half cycle

the object made four and half cycles in the anticlockwise directions

c) $\frac{7}{2}\pi=3.5\pi=2\pi+1.5\pi$, $1.5\pi$ is a three quarter cycle $(\frac{1.5\pi}{2\pi}=\frac{3}{4})$

the object made one and three quarter cycles in anticlockwise direction