Measurement of Angle: Radian and Degree

1.  To measure an angle in degree: The angle between two perpendicular lines is called a right angle. A right angle is equal to 90 degree, it is written as 90°.

Thus, if a right angle is divided into 90 equal parts then one part is called one degree. It is written as 1°.

If 1° is divided into 60 equal parts, each part is called 1 minute. It is denote by 1′.

\frac{1}{60}th part of 1′ is called one second. It is written as 1”.

Hence right angle = 90°
=90 \times 60 = 5400’=5400
=90 \times 60 \times 60 =324000=324000 seconds

Again,

1°=60’=60\times 60”=3600”

2. To measure an angle in radian: let AB be an arc of a given circle

whose length is equal to radius of the circle . The angle subtended by arc AB at the centre O of the circle is measured as 1 radian i.e, angle(AOB)  radian. It is denoted by 1 or 1 rad.

In the given figure,

It is also written as 1° .

3. Relation between degree measure and radian measure:

Thus to change degree into radian, multiply by \frac{\pi}{180} and to change radian into degree multiply by \frac{180}{\pi} . If mentioned, take \pi=\frac{22}{7} 0r 3.14.

Solved Example

1. Convert the following degree measures in the radian measure.

(i) 42°30′                                                     (ii) -520°

Solution:

2. Convert the following radian measure in degree measures

(i) 4                                                                  (ii) \frac{-5\pi}{3}

Solution: (i)

3. A wheel makes 180 revolutions in one minute. Through how many radians does it turn in one second? Also find its degree measure.

Solution:   Wheel makes 10 revolution in 60 seconds

Therefore, Wheel makes  \frac{180}{60}=3  revolution in 1 second.

Now,   As  one complete revolution measures  2\pi radian.

Therefore, three complete revolutions measure = 2\pi\times3=6\pi  radian

Again, As \pi rad = 180 ^{o}

Therefore, 6\pi rad = 6 \times180 ^{o}=1080 ^{o}

4. Find the degree and radian measure of the angle subtended at the centre of a circle of radius 200 cm by an arc of length 11 cm.

Solution:

5. In a circle of diameter 50 cm, the length of a chord is 25 cm. find the length of minor arc and major arc of the chord.

Solution: see the figure

 

6. If in two circles, arcs of the same length subtend angle 60° and 75° to the centre, find the ratio of their radii.

Solution: let the radii of two circles, be r1 and r2 respectively.

According to the question, arc AB=L  (say) in the two circle.

7. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc length 18 cm.

Solution: Suppose the pendulum swings through an angle of  \theta radian .

then, \theta=\frac{l}{r}=\frac{18}{75} rad (see figure) = \frac{6}{25} rad

8. Find the angle in radians between the hands of a clock at half past three.

Solution: In 60 minutes hand of a watch completes one revolution i.e., moves through an angle of  radian (360°).

Also, at three past half, the hour hand is exactly at the midway between 3 and 4, (shown by point A is figure) and minute hand is exactly at 6 (shown by point B in figure.)

Hence there is a difference of 2\times5+\frac{5}{2}=\frac{25}{2} minute between A and B.

Now, as 60 minute revolution = 2\pi \,rad

Therefore, \frac{25}{2} Minute revolution = \frac{25}{2\times60}(2\pi)=\frac{5\pi}{12} rad

Hence the two hands of the clock makes an angle of  \frac{5\pi}{12} rad at half past three.

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