Radians, as covered in a previous blog article on the unit circle, provide another way of angle measurement. Usually, when radians are first encountered, there has only been work done with angles measured in degrees. However, it is essential to become familiar with radians.

## What are Radians?

Radians are the SI unit for measuring angles and are the standard unit of angular measure used in many areas of mathematics.

SI is the International System of Units, which is the most widely used system of measurement and the only one with official status in almost all countries. It is used in science, tech, and industry.

Put simply, radians are like metric units in measurement. They are the main official means by which angles are measured. In mathematics, angles are measured along the number line in radians.

One radian is just under 57.3 degrees.

## Radian Unit Measure

Having grasped the importance of radians, it is necessary to know the symbol for them. It is well known that 60^{o} means 60 degrees, so how would you write 60 radians?

The answer is quite simple, 60 radians is written as either 60 rad or 60 c.

In mathematics, at advanced levels, it is taken for granted that radians are being used because radians are naturally along a number line. To see how this works with a unit circle have a look at the diagram below.

The angle is \dfrac{\pi}{3} in radians. In degrees, this angle is 60^{o}. We go over how we go from one angle measuring system to the other.

## Why do we use degrees?

The main reason is practical. \pi (Pi) is an irrational number, which means that it is impossible to write it as a whole number, or even a fraction, and 1 radian is a rather large angle of about 57.3^{o}.

Many professions, such as carpentry, plumbing and farming need a far more practical means of measuring angles and the system using degrees certainly provides this.

## Converting from degrees to radians and vice versa

The problem of angle conversion from one mode to the other is a simple exercise in proportion. The critical fact which makes this easy is that **180 ^{o} = **

**π radians**

*.*

**Example**

Change 85^{o} to radians.

*Answer*

180^{o} = \pi radians

Hence 1^{o} = \dfrac{\pi}{180} radians

Hence 85^{o} = \dfrac{85 \times \pi}{180} = 1.484 radians.

**Example**

Change 2.5 radians into degrees

*Answer*

π radians = 180^{o}

Hence 1 radian = \dfrac{180\degree}{\pi}

Hence 2.5 radians = \dfrac{2.5 \times 180\degree}{\pi} = 143.24\degree

Frequently it is necessary to express an angle in radians as a fraction of \pi . The following example shows this.

**Example**

Express 75^{o} as a fraction of \pi .

*Answer*

A requirement in many problems is that the answer is given as a fraction of \pi .

180^{o} = \pi radians

Hence 1^{o} = \dfrac{\pi}{180} \;radians

Hence 75^{o} = \dfrac{75 \pi}{180} \;radians

We can simplify this down to:

\dfrac{5}{12}Hence 75^{o} = \dfrac{5 \pi}{12} \;radians

## Exact angle values

There are some angles, which have values for trigonometric functions that are an exact fraction or surd.

For instance:

cos(\dfrac{\pi}{3}) = \dfrac{1}{3}sin(\dfrac{\pi}{3}) = \dfrac{\sqrt{3}}{2} .

The diagram below, shows why this is the case.

The triangle ABC is equilateral, so each of its angles is 60^{o}, which is \dfrac{\pi}{3} in radians.

AD is the perpendicular line from A to the opposite line in the triangle, which is BC.

By using Pythagoras’ theorem, it is easily proved that AD is \sqrt{3} in length.

sin(ACD) = sin(60\degree) = sin(\dfrac{\pi}{3} ) = \dfrac{opposite}{hypotenuse} = \dfrac{AD}{AC} = \dfrac{\sqrt{3}}{2}Similarly, cos(\dfrac{\pi}{3}) = \dfrac{1}{2} .

Another very important surd is that for the trigonometric functions of 45o or \dfrac{\pi}{4} radians.

Here is the proof of this useful result.

The diagram below of the unit circle may help with the understanding of this.

As the triangle has two equal angles, it is isosceles (has two equal sides), hence, cos(45^{o}) = sin(45^{o}).

From the theorem of Pythagoras, cos^{2}(45^{o}) + sin^{2}(45^{o}) = 1.

As cos(45^{o}) = sin(45^{o}), 2cos^{2}(45^{o}) = 1

Hence, cos2(45o) = \dfrac{1}{2} .

Hence, cos(45o) = \dfrac{1}{\sqrt{2}} and therefore sin(45o) = \dfrac{1}{\sqrt{2}} .

Hence, as 45o = \dfrac{\pi}{4} radians.

cos(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} radians and sin(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} radians

## Radian and Degree Angle Conversions

The following table gives some very useful conversions between radians and degrees and some frequently used surd values.

Angle x^{o} | Angle in radians | cos(x^{o}) | sin(x^{o}) | tan(x^{o}) = sin(x^{o})/ cos(x^{o}) |

0^{o} | 0 | 1 | 0 | 0 |

30^{o} | \dfrac{\pi}{6} | \dfrac{sqrt{3}}{2} | \dfrac{1}{2} | \dfrac{1}{\sqrt{3}} |

45^{o} | \dfrac{\pi}{4} | \dfrac{1}{\sqrt{2}} | \dfrac{1}{\sqrt{2}} | 1 |

60^{o} | \dfrac{\pi}{3} | \dfrac{1}{2} | \dfrac{sqrt{3}}{2} | \sqrt{3} |

90^{o} | \dfrac{\pi}{2} | 0 | 1 | Undefined |

If there is a need to extend values to the 2nd, 3rd, and 4th quadrants then methods, which are identical to those used with degrees are necessary.

To determine in which quadrants circular functions are positive, the mnemonic ** All Science Teachers Cr**y is very helpful. In this:

**A**stands for

**All**positive,

**S**stands for

**Sine**is positive,

**T**stands for

**Tangent**is positive and

**C**stands for

**Cosine**is positive. With this, circular functions of any angle can be used by using acute angles. Have a look at the diagram.

**How does this work?**

**Exampl**e **1**

Find the cosine, sine, and tangent of \dfrac{2\pi}{3} .

\dfrac{2\pi}{3} is a second quadrant angle.

In the second quadrant, the angle needs to be subtracted from *π* to get the acute angle, which will give the answers.

Hence, the acute angle is \dfrac{4\pi}{3} - \pi = \dfrac{\pi}{3} .

sin(\dfrac{4\pi}{3}) = -sin(\dfrac{\pi}{3}) = - \dfrac{sqrt{3}}{2} or -0.8660 (Only tangent is positive in quadrant 3)

cos(\dfrac{4\pi}{3}) = -cos(\dfrac{\pi}{3}) = 0.5 (Only tangent is positive in quadrant 3)

tan(\dfrac{4\pi}{3}) = tan(\dfrac{\pi}{3}) = \sqrt{3} or 1.732 (Only tangent is positive in quadrant 3)

**Exampl**e **2**

Find the cosine, sine, and tangent of \dfrac{5\pi}{3} .

\dfrac{5\pi}{3} is a fourth quadrant angle.

In the third quadrant, *π* needs to be subtracted from the angle to get the acute angle, which will give the answers.

Hence, the acute angle is 2\pi - \dfrac{5\pi}{3} = \dfrac{\pi}{3}

sin(\dfrac{5\pi}{3}) = -sin(\dfrac{\pi}{3})= -\dfrac{sqrt{3}}{2} or -0.8660 (Only cosine is positive in quadrant 4)

cos(\dfrac{5\pi}{3}) = cos(\dfrac{\pi}{3}) = 0.5 (Only cosine is positive in quadrant 4)

tan(\dfrac{5\pi}{3}) = -tan(\dfrac{\pi}{3}) = -\sqrt{3} or -1.732 (Only cosine is positive in quadrant 4)

## Negative Radian Angles

If you have a negative angle, then all you need to know is which quadrant you’re in.

A negative angle has the same values for circular functions as the negative angle added to 2π.

### Example

The following example shows this.

\dfrac{\pi}{3} is a fourth quadrant angle. We add -\dfrac{\pi}{3} to 2\pi , which gives \dfrac{5\pi}{3} .

In the fourth quadrant, the angle needs to be subtracted from 2\pi to get the acute angle, which will give the answers.

Hence, the acute angle is 2\pi - \dfrac{5\pi}{3} = \dfrac{\pi}{3}

sin(\dfrac{5\pi}{3}) = -sin(\dfrac{\pi}{3}) = -\dfrac{sqrt{3}}{2} or -0.8660 (Only cosine is positive in quadrant 4)

cos(\dfrac{5\pi}{3}) = cos(\dfrac{\pi}{3}) = 0.5 (Only cosine is positive in quadrant 4)

tan(\dfrac{5\pi}{3}) = -tan(\dfrac{\pi}{3}) = -\sqrt{3}or -1.732 (Only cosine is positive in quadrant 4)

## Radian Graphs

The graphs of trigonometric functions using radians are very similar to those when degrees are used.

We will graph the major trigonometric functions and list important properties, adjusted for radians.

## Cosine Graph

Using the table then plotting the graph or using one of the brilliant graph drawing applications on the Internet, such as Desmos, the following graph is obtained.

This graph shows some important properties of the cosine function:

(i) The cosine function does not take any value outside the range -1 < *y* < 1.

(ii) The cosine function repeats itself every 2\pi . If a function is repeating in this way, it is called periodic. The period of the cosine function is 2\pi . This is the smallest value that the cosine function repeats itself over.

As a consequence of this, for any angle x, cosx = cos(2\pi + x).

(iii)The cosine function’s values for negative angles are exactly the same as for the corresponding positive angle, ie cos*x* = cos(-*x*).

The effect on the graph for this is that the graph is symmetrical about the *y* axis. Functions which are symmetrical about the *y* axis are called ** even**.

(iv) For any angle θ, cosx = cos(2\pi – x) and cos(\pi - x) = cos(\pi + x). These properties are sometimes called the symmetry properties of the cosine function.

## Sine Graph

This graph shows some important properties of the sine function:

(i) The sine function does not take any value outside the range -1 < *y* < 1.

(ii) The sine function repeats itself every 2\pi . If a function is repeating in this way, it is called periodic. The period of the sine function is 2\pi . This is the smallest value that the sine function repeats itself over.

As a consequence of this, for any angle x, sinx = sin(2\pi + x).

(iii) The sine function’s values for negative angles are the negative of the corresponding positive angle, (ie sin(*x*) = sin(*x*)). The effect on the graph for this is that the graph has point symmetry about the origin (0,0). Such functions are called ** odd**.

(iv) For any angle x, sin x = sin(\pi – x). This is sometimes called the symmetry property of the sine function.

## Tangent Graph

This graph shows some important properties of the tangent function:

(i) The tangent function takes all values in the real numbers. Unlike cosine and sine, it is unbounded.

(ii) The tangent function repeats itself every \pi . If a function is repeating in this way, it is called periodic. The period of the tangent function is \pi . This is the smallest value that the sine function repeats itself over.

As a consequence of this, for any angle x, tanx = tan(\pi + x).

(iv) The tangent function’s values for negative angles are the negative of the corresponding positive angle, (ie tan(*x*) = tan(*x*)). The effect on the graph for this is that the graph has point symmetry about the origin (0,0). Such functions are called ** odd**.

Tangent and sine are both odd. Other well-known odd functions are *y* = *x* and *y* = *x*^{3}.