Angle Measurement: Degrees and Radians

Angle and Its Measurement Systems (Degree Measure)

An angle is a fundamental concept in geometry and trigonometry. It is formally defined as the measure of rotation of a ray about its fixed initial point. The initial position of the ray is called the initial side, and the final position after rotation is called the terminal side. The fixed point about which the ray rotates is called the vertex of the angle.

The direction of rotation determines whether the angle is positive or negative:

• If the rotation is in the anticlockwise (counterclockwise) direction, the angle is considered positive.

• If the rotation is in the clockwise direction, the angle is considered negative.

Angles can have any real value, positive or negative, including values greater than $360^\circ$ (or $2\pi$ radians), representing multiple rotations.

There are several systems for measuring angles. The two most common systems are the Degree measure and the Radian measure.

Degree Measure (Sexagesimal System)

The degree measure is one of the oldest systems for measuring angles and is widely used in practical fields like surveying, engineering drawings, and navigation. It is also the primary system introduced in basic geometry and trigonometry.

Definition: In the degree system, a complete rotation of a ray back to its original position is divided into 360 equal parts. Each part is called one degree. The symbol for a degree is $^\circ$.

So, the measure of a complete angle (one full revolution) is $360^\circ$.

For finer measurements, a degree is further subdivided:

• Each degree is divided into 60 equal parts called minutes. The symbol for a minute is $'$.

  $1^\circ = 60'$

• Each minute is divided into 60 equal parts called seconds. The symbol for a second is $''$.

  $1' = 60''$

From these definitions, it follows that:

$1^\circ = 60' = 60 \times 60'' = 3600''$

Due to the division into 60 parts at each level of subdivision (except for the initial division of the full circle), this system is also known as the Sexagesimal system (from Latin 'sexagesimus', meaning sixtieth).

Examples of angles in degrees:

• A right angle, representing one-quarter of a full revolution, measures $ \frac{1}{4} \times 360^\circ = 90^\circ $. An angle measuring less than $90^\circ$ is called an acute angle. An angle measuring greater than $90^\circ$ but less than $180^\circ$ is called an obtuse angle.

• A straight angle, representing half a revolution, measures $ \frac{1}{2} \times 360^\circ = 180^\circ $.

• Common angles like $30^\circ, 45^\circ, 60^\circ$ are frequently encountered acute angles.

• Angles can exceed $360^\circ$. For example, an angle of $420^\circ$ represents one complete revolution ($360^\circ$) plus an additional $60^\circ$. Similarly, a negative angle like $-90^\circ$ represents a quarter revolution in the clockwise direction.

Angles in degree measure are often written with fractional parts as decimals (e.g., $45.5^\circ$) or in the degree-minute-second (DMS) format (e.g., $45^\circ 30'$). For instance, $45.5^\circ = 45^\circ + 0.5^\circ = 45^\circ + 0.5 \times 60' = 45^\circ 30'$.

Example 1: Converting between Decimal Degrees and DMS

Answer:

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Radian Measure

The radian measure is the standard unit of angular measurement in many areas of mathematics (especially calculus), physics, and engineering. It is defined based on the properties of a circle.

Definition of a Radian

Consider a circle with its center at the origin and a radius of length $r$. A radian is defined as the measure of the angle subtended at the center of the circle by an arc whose length is exactly equal to the radius of the circle.

In the figure, if the length of the arc AB ($l$) is equal to the radius OA ($r$), then the angle $\angle AOB$ is defined as 1 radian.

More generally, if an arc of length $l$ subtends an angle $\theta$ (in radians) at the center of a circle of radius $r$, the relationship between the arc length, radius, and the angle is given by:

$\mathbf{\theta (\text{in radians}) = \frac{\text{Length of Arc}}{\text{Radius}} = \frac{l}{r}}$

This formula can be rearranged to find the arc length: $ l = r\theta $. It's crucial that $\theta$ is in radians for this formula to be valid.

Since $\theta$ is a ratio of two lengths, its unit (radian) is dimensionless. When an angle is given without an explicit unit ($^\circ$), it is usually assumed to be in radians (e.g., $\sin 2$ implies $\sin(2 \text{ radians})$).

Relationship between Radian and Degree Measure

The fundamental link between radians and degrees comes from considering the angle subtended by the entire circumference of the circle at the center.

• In degree measure, the angle for a full circle is $360^\circ$.

• The length of the entire circumference of a circle with radius $r$ is $l = 2\pi r$.

Using the radian definition $\theta = \frac{l}{r}$, the angle subtended by the circumference in radians is:

$ \theta_{\text{full circle}} = \frac{2\pi r}{r} = 2\pi $ radians.

Since both $360^\circ$ and $2\pi$ radians represent one full revolution, they are equivalent:

Dividing both sides by 2, we get the most commonly used conversion relationship:

This equation is the basis for converting between the two systems.

Value of 1 Radian in Degrees

From the relationship $\pi$ radians = $180^\circ$ (Equation 4.2), we can find the degree measure equivalent to 1 radian:

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

To get a numerical value, we use an approximate value for $\pi$. Using $\pi \approx \frac{22}{7}$:

$ 1 \text{ radian} \approx \frac{180^\circ}{22/7} = \frac{180 \times 7}{22}^\circ = \frac{90 \times 7}{11}^\circ = \frac{630}{11}^\circ $

Let's convert this fraction to degrees, minutes, and seconds:

$ \frac{630}{11} = 57 \text{ with a remainder of } 3 $. So, $ \frac{630}{11}^\circ = 57 \frac{3}{11}^\circ $.

$ 57 \frac{3}{11}^\circ = 57^\circ + \frac{3}{11}^\circ $

Convert the fractional part of the degree to minutes ($1^\circ = 60'$):

$ \frac{3}{11}^\circ = \left(\frac{3}{11} \times 60\right)' = \frac{180}{11}' $

$ \frac{180}{11} = 16 \text{ with a remainder of } 4 $. So, $ \frac{180}{11}' = 16 \frac{4}{11}' $.

$ 16 \frac{4}{11}' = 16' + \frac{4}{11}' $

Convert the fractional part of the minute to seconds ($1' = 60''$):

$ \frac{4}{11}' = \left(\frac{4}{11} \times 60\right)'' = \frac{240}{11}'' $

$ \frac{240}{11} \approx 21.81...'' $ which is approximately $22''$ when rounded to the nearest second.

So, $1 \text{ radian} \approx 57^\circ 16' 22''$ (approximately).

Using a more precise value of $\pi \approx 3.14159$, $1 \text{ radian} \approx \frac{180^\circ}{3.14159} \approx 57.296^\circ$.

Value of 1 Degree in Radians

From the relationship $180^\circ = \pi$ radians (Equation 4.2), we can find the radian measure equivalent to 1 degree:

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

Using an approximate value of $\pi \approx 3.14159$:

$ 1^\circ \approx \frac{3.14159}{180} \approx 0.01745 $ radians (approximately).

So, $1^\circ \approx 0.01745 \text{ radians}$ (approximately).

Common Angles in Radians

It is useful to remember the radian measures for some common angles:

• $ 30^\circ = 180^\circ / 6 = \pi / 6 $ radians

• $ 45^\circ = 180^\circ / 4 = \pi / 4 $ radians

• $ 60^\circ = 180^\circ / 3 = \pi / 3 $ radians

• $ 90^\circ = 180^\circ / 2 = \pi / 2 $ radians

• $ 180^\circ = \pi $ radians

• $ 270^\circ = 3 \times 90^\circ = 3 \times \pi / 2 = 3\pi / 2 $ radians

• $ 360^\circ = 2 \times 180^\circ = 2\pi $ radians

Conversion between Degree and Radian Measure

As we have learned about the two common systems for measuring angles, Degree measure and Radian measure, it is essential to be able to convert an angle from one system to the other. The conversion is based on the fundamental relationship between these two units, which we derived from considering a full circle:

This single relationship allows us to create conversion factors for both directions.

1. Converting Degrees to Radians

To convert an angle measured in degrees to its equivalent measure in radians, we use the fact that $180^\circ = \pi$ radians. Dividing both sides by $180^\circ$, we get the conversion factor: $ 1^\circ = \frac{\pi}{180} \text{ radians} $.

Therefore, to convert a degree measure to a radian measure, we multiply the degree measure by $\frac{\pi}{180}$.

Formula:

Radian Measure = Degree Measure $ \times \frac{\pi}{180} $

Examples: Converting Degrees to Radians

Answer:

2. Converting Radians to Degrees

To convert an angle measured in radians to its equivalent measure in degrees, we again use the relationship $\pi \text{ radians} = 180^\circ$. Dividing both sides by $\pi$, we get the conversion factor: $ 1 \text{ radian} = \frac{180^\circ}{\pi} $.

Therefore, to convert a radian measure to a degree measure, we multiply the radian measure by $\frac{180^\circ}{\pi}$.

Formula:

Degree Measure = Radian Measure $ \times \frac{180^\circ}{\pi} $

Examples: Converting Radians to Degrees

Answer:

Common Angle Conversions

It is highly recommended to memorise the radian equivalents of common degree measures, as they appear frequently in trigonometric problems.

Note for Competitive Exams

Length of an Arc and Area of a Sector (using Radian Measure)

One of the main reasons why radian measure is preferred in higher mathematics and applications is that it significantly simplifies formulas related to circles, particularly the length of an arc and the area of a sector. These formulas become elegant and easy to use when the angle is measured in radians.

Consider a circle with radius $r$ and center O. Let a central angle of measure $\theta$ intercept an arc AB on the circle.

The region OAB, bounded by the two radii OA, OB and the arc AB, is called a sector of the circle. The length of the curved boundary AB is called the arc length, denoted by $s$.

1. Length of an Arc ($s$)

The formula for the length of an arc is derived directly from the definition of radian measure. The definition states that the radian measure of a central angle is the ratio of the length of the intercepted arc to the radius of the circle:

Let the arc length be $s$ and the radius be $r$. Then,

Rearranging this formula to solve for $s$, we get the formula for the length of a circular arc:

Formula for Arc Length:

$s = r \theta$

where:

• $s$ is the length of the arc

• $r$ is the radius of the circle

• $\theta$ is the central angle subtended by the arc, and it MUST be measured in radians.

Example 1: Calculating Arc Length

Answer:

2. Area of a Sector ($A$)

A circular sector is like a slice of pizza or pie. Its area is a fraction of the total area of the circle, determined by the central angle of the sector.

The area of the whole circle (which corresponds to a central angle of $2\pi$ radians) is $ \pi r^2 $. The area of a sector with central angle $\theta$ (in radians) is proportional to the ratio of $\theta$ to the total angle of the circle ($2\pi$).

Using proportionality:

$ \frac{A}{\pi r^2} = \frac{\theta}{2\pi} $

Solving for $A$:

$ A = \frac{\theta}{2\pi} \times \pi r^2 $

$ A = \frac{\theta \times \cancel{\pi} r^2}{2\cancel{\pi}} $

$ A = \frac{1}{2} r^2 \theta $

Formula for Area of a Sector:

$A = \frac{1}{2} r^2 \theta$

where:

• $A$ is the area of the sector

• $r$ is the radius of the circle

• $\theta$ is the central angle of the sector, and it MUST be measured in radians.

Alternative Formula using Arc Length:

We know that $s = r\theta$. From this, we can write $\theta = \frac{s}{r}$. Substituting this into the area formula $A = \frac{1}{2} r^2 \theta$:

$ A = \frac{1}{2} r^2 \left(\frac{s}{r}\right) $

$ A = \frac{1}{2} \frac{r^2 s}{r} $

Assuming $r \ne 0$, we can cancel one $r$ from the numerator:

$ A = \frac{1}{2} \frac{r^{\cancel{2}} s}{\cancel{r}} $

$A = \frac{1}{2} rs$

This formula is useful if you know the radius and the arc length but not the central angle (or if you want to calculate area after finding arc length).

Example 2: Calculating Area of a Sector

Answer: