Measuring angles

An angle is the region of a plane formed between two rays that share a common starting point, denoted as \( A \).

To measure an angle, you need to choose a reference unit and assign it a numerical value based on its size.

This unit must follow some fundamental principles:

• Two angles are congruent if and only if they have the same measure.
• If one angle is a multiple of another, their numerical values must maintain the same proportion.
• The measure of the sum of two angles is equal to the sum of their individual measures.

Once a unit of measurement is defined, you can use it to determine an angle’s size.

An angle’s measure is called its magnitude, and there are different units used to express it. The most common ones are degrees (sexagesimal system) and radians.

Example: Suppose you have two angles—one measuring 30° and the other 60°. Since 60° is exactly twice 30°, you can say the second angle is twice the first, maintaining proportionality. Moreover, adding these two angles together gives you 90°, forming a right angle. This example illustrates how measuring angles in degrees follows all the fundamental principles we discussed.

The Sexagesimal System

The most widely used system for measuring angles is the sexagesimal system, introduced by Babylonian astronomers.

In this system, the primary unit is the degree, represented by the symbol °.

A full rotation, meaning a complete turn around a point, is divided into 360 degrees.

Key angle measures in degrees include:

• Right angle: 90°
• Straight angle: 180°
• Full rotation: 360°

For greater precision, a degree is further divided into smaller units:

• Minute ('): One-sixtieth of a degree → \(1' = \frac{1}{60}^\circ\).
• Second (''): One-sixtieth of a minute → \(1'' = \frac{1}{60}' = \frac{1}{3600}^\circ\).

Today, instead of minutes and seconds, angles are often expressed in decimal form to simplify calculations, especially when using electronic devices.

Example

Imagine looking at an analog clock showing 3:10:30 and wanting to determine the exact angle between the hour and minute hands, expressed in degrees, minutes, and seconds.

How much has the hour hand moved?

The hour hand completes a full 360° rotation in 12 hours, meaning it moves 30° per hour.

$$ 360° \div 12 = 30° $$

In one minute, it moves 0.5°.

To find this, divide 30° by 60, since there are 60 minutes in an hour.

$$ 30° \div 60 = 0.5° $$

In one second, it moves 0.0083°.

Following the same logic, divide 0.5° by 60, as there are 60 seconds in a minute.

$$ 0.5° \div 60 = 0.0083° $$

Now, let’s calculate the total movement from 12:00:00 to 3:10:30:

Each hour, the hour hand moves 30°.

$$ 30° \times 3 = 90° $$

In 10 minutes, it moves:

$$ 10 \times 0.5° = 5° $$

In 30 seconds, it moves:

$$ 30 \times 0.0083° = 0.249° $$

So, by 3:10:30, the hour hand is at:

$$ 90° + 5° + 0.249° = 95.249° $$

How much has the minute hand moved?

The minute hand completes a full 360° rotation in 60 minutes, so it moves 6° per minute.

$$ 360° \div 60 = 6° $$

In one second, it moves 0.1°.

$$ 6° \div 60 = 0.1° $$

Now, let’s determine its position at 10 minutes and 30 seconds.

In 10 minutes, it moves:

$$ 10 \times 6° = 60° $$

In 30 seconds, it moves:

$$ 30 \times 0.1° = 3° $$

So, by 3:10:30, the minute hand is at $ 63° $

What is the angle between the two hands?

To find out, take the absolute difference between their positions:

$$ |95.249° - 63°| = 32.249° $$

So, the angle between the hour and minute hands at 3:10:30 is approximately 32.249°.

Converting to degrees, minutes, and seconds

The whole number part of 32.249° remains 32°.

To convert the decimal part (0.249°) into minutes:

$$ 0.249 \times 60 = 14.94' $$

Since we only take the whole number, that gives us 14 minutes.

Now, converting the remaining decimal (0.94') into seconds:

$$ 0.94 \times 60 = 56.4'' $$

Thus, the angle between the hour and minute hands at 3:10:30 is approximately 32° 14' 56'' (thirty-two degrees, fourteen minutes, and fifty-six seconds).

Degrees vs. Radians

Aside from degrees, another common unit for measuring angles is the radian. This system is based on the relationship between an angle and the arc length it subtends on a unit circle.

In this system, a full revolution (360°) is equal to \(2\pi\) radians because the circumference of a unit circle is \(2\pi\).

Thus, a straight angle (180°) is \(\pi\) radians, and a right angle (90°) is \(\frac{\pi}{2}\) radians.

The advantage of radians is that they simplify many calculations in trigonometry and mathematical analysis, making it easier to work with trigonometric functions and their derivatives.

Converting Between Degrees and Radians

You can convert between degrees and radians using this formula:

$$ 360° = 2 \pi \text{ rad} $$

Dividing both sides by 360 gives:

$$ 1^\circ = \frac{\pi}{180} \text{ rad} $$

Similarly, to convert radians to degrees:

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

This means that \(180^\circ = \pi\) radians, \(90^\circ = \frac{\pi}{2}\) radians, and \(45^\circ = \frac{\pi}{4}\) radians.

For example, the clock angle of 32.249° converts to radians as follows: $$ 32.249° \times \frac{\pi}{180°} \approx 0.56 \text{ rad} $$

Which System Should You Use?

Degrees are more common in everyday applications like navigation and cartography, while radians are preferred in theoretical fields such as physics and calculus due to their advantages in trigonometric calculations.