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′.
part of 1′ is called one second. It is written as 1”.
|Hence right angle = 90°
=90 60 = 5400’=5400
=90 60 60 =324000=324000 seconds
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 and to change radian into degree multiply by . If mentioned, take 0r 3.14.
1. Convert the following degree measures in the radian measure.
(i) 42°30′ (ii) -520°
2. Convert the following radian measure in degree measures
(i) 4 (ii)
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 revolution in 1 second.
Now, As one complete revolution measures radian.
Therefore, three complete revolutions measure = radian
Again, As rad =
Therefore, 6 rad = 6
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.
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 radian .
then, rad (see figure) = 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 minute between A and B.
Now, as 60 minute revolution =
Therefore, Minute revolution = rad
Hence the two hands of the clock makes an angle of rad at half past three.