Chapter 3.2 - Measurement of Angles using Radians

In the previous section, we saw the definition of degree measure. In this section, we will see radian measure.

Radian measure

This can be written in 8 steps:
1. Consider a circle of radius 1 meter.
    ♦ The circumference of that circle will be 2π.
    ♦ This 2π is approximately equal to (2 × 3.14) = 6.28 m.
• That means, the circumference of the circle will be approximately equal to 6.28 m.
2. This 6.28 meter is greater than 1 meter. So we can easily mark an arc on the circle in such a way that, the length of that arc is 1 meter. This is shown in fig.3.6 below.
• Arc AB has a length of 1 meter.

Fig.3.6

3. Next step is to draw the two radii OA and OB.
• Now we have a sector formed between the three items: radius OA, radius OB and arc AB.
◼ The central angle of this sector is said to be one radian.
   ♦ Using symbols, it is written as 1^c
4. Based on this, we can write any other angle in radian measure. Let us see some examples:
◼ In fig.b, the ray rotates in such a way that, length of the arc AB is 1¹⁄₂ meter. Then the amount of rotation in radian is 1¹⁄₂^c.
• The rotation is in the anticlockwise direction. So the angle is +1¹⁄₂^c. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 1¹⁄₂^c.    
◼ In fig.c, the ray rotates in such a way that, length of the arc AB is 2 meter. Then the amount of rotation in radian is 2^c.
• The rotation is in the anticlockwise direction. So the angle is +2^c. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 2^c.
◼ In fig.d, the ray rotates in such a way that, length of the arc AB is 1 meter. Then the amount of rotation in radian is 1^c.
• The rotation is in the clockwise direction. So we must write -1^c.
◼ In fig.e, the ray rotates in such a way that, length of the arc AB is 1¹⁄₂ meter. Then the amount of rotation in radian is 1¹⁄₂^c.
• The rotation is in the clockwise direction. So we must write -1¹⁄₂^c.
◼ In fig.f, the ray rotates in such a way that, length of the arc AB is 2 meter. Then the amount of rotation in radian is 2^c.
• The rotation is in the clockwise direction. So we must write -2^c.
5. Now we will try to obtain a general form for writing any angle in radian measure. The following steps (6), (7) and (8) will help us to obtain the general form.
6. Let us consider the case when the ray rotates through one fourth of a revolution. It can be written in 6 steps:
(i) In fig.3.7(a) below, rotation of a ray OP is shown. Initial side is OP and terminal side is OQ.

Fig.3.7

(ii) The rotation is exactly one fourth of a revolution.
• So the length of arc PQ = l meter = $\frac{1}{4}\times 2\pi r$
    ♦ Where r is the radius of the outer circle.
    ♦ OP = OQ = r
(iii) The green circle is a concentric circle. It has a radius of 1 m.
• This circle cuts OP and OQ at A and B
• So the length of arc AB = l₁ = $\frac{1}{4}\times 2\pi \times 1$
(iv) Thus we get: $\frac{l_1}{l}=\frac{\frac{1}{4}\times 2\pi \times 1}{\frac{1}{4}\times 2\pi r}=\frac{1}{r}$
• This can be rearranged as: $l_1=\frac{l}{r}$
(v) Here l₁ is the 'length of arc (of the unit circle) between OA and OB'.
• But based on fig.3.6, 'length of arc (of the unit circle) between OA and OB' is the angle (in radians) between OA and OB.
• Also, the angle between OA and OB is same as the angle between OP and OQ.
(vi) Based on (iv) and (v), we can write:
Angle between OP and OQ = $\frac{l}{r}$
   ♦ Where l is the length of arc PQ and r is the radius of that arc.
7. Let us check the above result for another angle. It can be written in 6 steps:
(i) In fig.3.7(b) above, rotation of a ray OP is shown. Initial side is OP and terminal side is OQ.
(ii) The rotation is exactly one eighth of a revolution.
• So the length of arc PQ = l meter = $\frac{1}{8}\times 2\pi r$
    ♦ Where r is the radius of the outer circle.
    ♦ OP = OQ = r
(iii) The green circle is a concentric circle. It has a radius of 1 m.
• This circle cuts OP and OQ at A and B
• So the length of arc AB = l₁ = $\frac{1}{8}\times 2\pi \times 1$
(iv) Thus we get: $\frac{l_1}{l}=\frac{\frac{1}{8}\times 2\pi \times 1}{\frac{1}{8}\times 2\pi r}=\frac{1}{r}$
• This can be rearranged as: $l_1=\frac{l}{r}$
(v) Here l₁ is the 'length of arc (of the unit circle) between OA and OB'.
• But based on fig.3.6, 'length of arc (of the unit circle) between OA and OB' is the angle (in radians) between OA and OB.
• Also, the angle between OA and OB is same as the angle between OP and OQ.
(vi) Based on (iv) and (v), we can write:
Angle between OP and OQ = $\frac{l}{r}$
   ♦ Where l is the length of arc PQ and r is the radius of that arc.
8. We get the same result in (6)and (7). This is because, whatever the 'fraction of revolution' be, it will be cancelled from the numerator and denominator. The final result will be the same.
• So we can use this result to find the angle between any two initial and terminal sides. It can be explained in 8 steps:
(i) In fig.3.8(a) below, OP and OQ are any two initial and terminal sides. We want to write the angle between them at O.

Fig.3.8

(ii) For that, we draw a unit circle with center at O. This circle is cut by OP and OQ at A and B respectively. This is shown in fig.b. So the required arc from the unit circle is AB
(iii) The length of arc AB will be $\frac{l}{r}$
   ♦ Where l is the length of arc PQ and r is the radius of that arc.
• But based on fig.3.6, 'length of arc (of the unit circle) between OA and OB' is the angle (in radians) between OA and OB.
• Also, the angle between OA and OB is same as the angle between OP and OQ.
(vi) So we can write:
Eq.3.1: Angle (in radians) between OP and OQ = $\frac{l}{r}$.
   ♦ Where l is the length of arc PQ and r is the radius of that arc.
• This can be written in simple words as:
$\mathbf\small{\text{Angle (in radians)}=\frac{\text{Arc}}{\text{radius}}}$
(vii) Once we understand the basics, we will no longer need to draw the unit circle. We can straight away write:
Angle between OP and OQ = $\frac{l}{r}$ radians.
• This is because, the unit circle is not required to determine l and r.
(viii) In Eq.3.1 above, note that, we have the quantity 'length' in both numerator and denominator. So radian is just a number.
• However, we must be careful to use the same units in both numerator and denominator. For example, if l is written in cm, r must also be written in cm.

Relation between radian and real numbers

This can be written in 6 steps:
1. In fig.3.9 below, a real number line is drawn in vertical position. It is kept tangential to a unit circle.

Fig.3.9

• The unit circle touches the zero of the number line at A.
2. Imagine that, the number line is made of a rope.
• Then we can wind the number line around the unit circle.
   ♦ See dictionary meaning of 'wind' here.
   ♦ A video can be seen here.
• The positive upper part should be wound in the anticlockwise direction as indicated by the upper white curved arrow.    
• The negative lower part should be wound in the clockwise direction as indicated by the lower white curved arrow.
3. We know that, the real number line will have all types of numbers like √2, √3, √5, π, 3.3333… etc.,
• When we wind the rope in this way, every real number (positive as well as negative) will get a unique position on the circumference of the unit circle.
4. We know that, the number line can extend upto infinity in both directions. How can we accommodate all those numbers on the unit circle?
• For that, we assume that the thickness of the rope is infinitesimal (extremely small). Then whatever be the number of windings, the radius of the unit circle will be always 1 meter. Thus we can accommodate all the numbers.
5. What is the advantage of winding the number line around a unit circle?
The answer can be written in steps:
(i) In fig.3.9(b), OA is the initial side and OB is the terminal side. We want the angle of rotation.
(ii) Note down the readings at A and B. From those readings, the length of arc AB can be easily determined.
(iii) But the length of arc AB is nothing but the required angle in radians.
(iv) So we can write: The distances on the number line are actually angles in radians.
6. We can write another interesting point. It can be written in 4 steps:
(i) In fig.3.9(b), OA is the initial side and OB is the terminal side. Suppose that, in the fig.b, the initial side is horizontal. That is., point A coincides with zero of the number line.
(ii) Then the length of arc AB
= Reading at B – Reading at A
= Reading at B – 0
= Reading at B
(iii) That means, if the ray starts to rotate from the horizontal position, the angle of rotation will be equal to the reading at B.
(iv) The reading at B is a number.
◼ So we can write:
If the ray starts to rotate from the horizontal position, every number on the number line will indicate an angle of rotation.

---

So we have learned about degree and radian measures. In the next section, we will see the relation between those two.

Previous

Contents

Next

Copyright©2021 Higher secondary mathematics.blogspot.com