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Introduction to Radians

Neil Trivedi

Teacher

Neil Trivedi

Contents

Radians

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Practice Questions

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Radians

A radian, denoted by superscript , is a way to measure angles, defined as the ratio between a circle’s arc length and its radius. In other words, the angle  subtended at the centre by the arc tells us how many times larger the arc length is compared to the radius .

Radians are particularly useful in calculus and trigonometry because, unlike degrees, which have a unit of measurement, radians are unitless.

A circle is shown with centre 𝑂, radii 𝑂𝐴 and 𝑂𝐵 forming angle 𝜃, and the arc 𝐴𝐵 labelled 𝑙=𝑟𝜃 to illustrate the arc length formula.

For instance, radian is the angle subtended at the centre of the circle by an arc whose length is equal to the radius.

A circle is shown with centre 𝑂, radius 𝑟, and an arc 𝐴𝐵 subtended by an angle of 1 radian, illustrating that the arc length 𝑙 equals the radius 𝑟.

The arc length of a full circle is  (i.e., the circumference), so the angle subtended at the centre is   radians, which is equal to . Similarly, the arc length of a semi-circle is , meaning the angle subtended at the centre is  radians, which is equal to .

Diagrams of a semicircle and a full circle show how arc length depends on radius, with the semicircle labelled 𝑙=𝜋𝑟 and the full circle labelled 𝑙=2𝜋𝑟.

We can derive the conversion formulas between radians and degrees by determining the values of and  using the fact that .

and

Conversion between Radians and Degrees

Degrees Radians

Example 1:

Convert these angles to radians.

a) 

Single Step: To convert from degrees to radians, we multiply by  .

b) 

Single Step: To convert from degrees to radians, we multiply by  .

c) 

Single Step: To convert from degrees to radians, we multiply by .

No answer provided.

Example 2:

Convert these angles to degrees:

a) 

Single Step: To convert from degrees to radians, we multiply by  .


b) 


Single Step: To convert from degrees to radians, we multiply by  .


c) radians


Single Step: To convert from degrees to radians, we multiply by  .


No answer provided.

Common Angles to become familiar with

Here are the sine, cosine, and tangent graphs in radians, in the interval .

The standard sine curve 𝑦=sin𝑥 is plotted from 𝑥=0 to 𝑥=2𝜋, showing a maximum at 𝜋/2, zeros at 0, 𝜋, and 2𝜋, and a minimum at 3𝜋/2.

No answer provided.

A sine curve is shown on 0≤𝑥≤2𝜋, rising from 0 to a maximum of 1 at 𝑥=𝜋 and returning to 0 at 𝑥=2𝜋, illustrating the stretched function 𝑦=sin(𝑥2).

No answer provided.

The graph of 𝑦=tan𝑥 is shown from 0 to 2𝜋, with vertical asymptotes at 𝑥=𝜋/2 and 𝑥=3𝜋/2 and zeros at 0, 𝜋, and 2𝜋.

No answer provided.

Example 3:

Sketch, in the interval , the graph of .

Step 1: Since the sine function takes  as its argument, we modify the range by dividing the values by .


Step 2: Sketch the graph of  between the modified range  and .

The graph of 𝑦=sin𝑥 is shown on 0≤𝑥≤𝜋, rising from 0 to a maximum of 1 at 𝑥=𝜋/2 and returning to 0 at 𝑥=𝜋.

Step 3: Un-modify the range by multiplying all values by . This enlarges the graph parallel to the  axis by a scale factor of , giving us the  graph in the range .A sine curve is shown on 0≤𝑥≤2𝜋, rising from 0 to a maximum of 1 at 𝑥=𝜋 and returning to 0 at 𝑥=2𝜋, illustrating the stretched function 𝑦=sin(𝑥/2).

No answer provided.

Example 4:

Sketch, in the interval , the graph of .

Step 1: Since the tan function takes  as its argument, we modify the range by adding  to the  values. 

Step 2: Sketch the graph of between  and .The graph of 𝑦=tan𝑥 is shown over an extended range of 𝑥, with repeating branches separated by vertical asymptotes at odd multiples of 𝜋/2.

Step 3: Un-modify the range by subtracting  from all  values. This shifts the graph, including the asymptotes, left by , resulting in the graph of  in the range  .

The graph of 𝑦=tan(2𝑥) is shown from 0 to 2𝜋, with more frequent vertical asymptotes and repeating branches to illustrate the reduced period compared with 𝑦=tan𝑥.

No answer provided.

Practice Questions

Further Practice Questions

Arcs, Sectors and Segments