In the previous section,
we saw some simple trigonometric identities. In this section, we
will see angle measurements using degrees and radians. First we will see the definition of angle.
Definition of angle
This can be written in 3 steps:
1. In fig.3.2(a) below, a ray OA is shown. O is the initial point of the ray.
 |
| Fig.3.2 |
• The ray is rotated in the anticlockwise direction, with O as center.
• OB is the final position of the ray after rotation.
2. We want to write the amount of rotation performed by the ray. To write that amount of rotation, we use the quantity called angle.
• So angle is the measure of the amount of rotation of a given ray about it’s initial point.
♦ Initial position of the ray is called initial side of the angle.
♦ Final position of the ray is called terminal side of the angle.
♦ The point of rotation is called vertex.
• In fig.b, we see that the angle is more. This is because, OA performs a greater amount of rotation.
3. In fig.c, we see that, OA performs a rotation in the clockwise direction.
• In fig.d, also the ray performs a rotation in the clockwise direction. But here the amount of rotation is more. So the angle is more.
4. Positive and negative angles:
• If the direction of rotation is anticlockwise, the angle is said to be positive.
• If the direction of rotation is clockwise, the angle is said to be negative.
Units for angles
• We know that, to measure a quantity, we need a unit. For example, to measure length, we use the unit meter. In some situations, instead of meter, the unit inch is used to measure length.
• In the same way, to measure angle, different units are available. For our present discussion, we consider three units:
♦ Revolutions per second
♦ Degree measure
♦ Radian measure
• Let us see each of them in detail:
Revolutions per second
This can be explained in 3 steps:
1. Consider the ray OA in fig.3.3(a) below. It is rotating about the point O.
• The point A moves along the dotted circle. OB is an intermediate position.
 |
| Fig.3.3 |
2. In fig.3.3(b), the ray has reached the final position. We see that, initial side and terminal side are the same.
• In such a situation, we say that, the ray performed one complete revolution.
3. This gives us a method to establish a unit to measure angle. It can be explained in 5 steps.
(i) Consider a wheel which is spinning rapidly. We want to write the amount of rotation performed by the wheel.
(ii) For that, draw a ray OA, where O is the center of the wheel and A is any convenient point on the circumference of the wheel.
(iii) Note the initial position of OA.
(iv) Due to the spin. The position of OA will be continuously changing.
• When OA reaches back at it’s initial position, we can say that, OA has completed one revolution.
• Obviously, the wheel also will have completed one revolution.
• If the wheel is spinning rapidly, it will be completing many revolutions in one second.
(v) This gives us a convenient method to write the 'amount of rotation' performed by a wheel.
• If OA reaches back to it's initial position 'n' times in one second, we can write: The wheel performs ‘n’ revolution per second.
Degree measure
This can be explained in 6 steps:
1. In fig.3.4(a) below, a circle is divided into 9 equal sectors.
 |
| Fig.3.4 |
• We have learned about sectors in our previous classes. (Details here). A sector is the area enclosed by an arc and two radii.
2. Imagine that, a circle is divided into 360 equal sectors.
• Consider any one of those sectors. That sector will have two radii.
• Let one of those two radii be the initial side OA of the ray. Let the other radius be the terminal side OB of the ray.
◼ Then the amount of rotation (angle) performed by the ray is said to be one degree.
♦ Using symbols, it is written as 1o.
3. We can write:
If a circle is divided into 360 equal sectors, the central angle of each of those sectors will be 1o.
4. We can see that, if the ray sweeps through all the 360 sectors, it will have completed one revolution.
• Also, if the ray sweeps through all the 360 sectors, it will have rotated through 360o. So one revolution is 360o.
◼ Based on this, we can write:
If the rotation from initial side to terminal side is 1⁄360 of a revolution, the angle is said to have a measure of 1o.
5. For precise scientific and engineering works, we will need to measure angles smaller than 1o.
• For that, we take any one of the 360 equal sectors. Then we divide that sector into 60 equal sectors. The central angle of each of those 60 sectors is said to be one minute. Using symbols, it is written as 1’.
◼ We can write: 1' = 1⁄60 of a degree. Or 1o = 60'
• Again take one of those 60 equal sectors. Divide that sector into 60 equal sectors. The central angle of each of those smaller 60 sectors is said to be one second. Using symbols, it is written as 1’’.
◼ We can write: 1'' = 1⁄60 of a minute. Or 1' = 60''.
6. Fig.3.5 below shows some examples where amount of rotation of the ray is measured in degrees.
 |
| Fig.3.5 |
◼ In fig.a, the ray completes one full revolution and reaches the initial position. So the amount of rotation in degrees is 360o.
• The rotation is in the anticlockwise direction. So the angle is +360o. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 360o.
◼ In fig.b, the ray completes one half revolution. So the amount of rotation in degrees is 360⁄2 = 180o.
• The rotation is in the anticlockwise direction. So the angle is +180o. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 180o.
◼ In fig.c, the ray completes three fourth revolution. So the amount of rotation in degrees is 360 × 3⁄4 = 270o.
• The rotation is in the anticlockwise direction. So the angle is +270o. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 270o.
◼ In fig.d, first, the ray completes one full revolution and reaches the initial position. But after reaching the initial position, it rotates further by an amount of 60o. So the amount of rotation in degrees is (360+60) = 420o.
• The rotation is in the anticlockwise direction. So the angle is +420o. For positive angle measures, the ‘+’ sign need not be shown. We can simply write 420o.
◼ In fig.e, the ray rotates in the clockwise direction. The amount of rotation is 30o. • Since the rotation is in the clockwise direction, we must write -30o.
◼ In fig.f, first, the ray completes one full revolution in the clockwise direction and reaches the initial position. But after reaching the initial position, it rotates further by an amount of 60o. So the amount of rotation in degrees is (360+60) = 420o.
• Since the rotation is in the clockwise direction, we must write -420o.
In
the next
section, we will see radian measure.
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