Angular measurement Objectives Be able to define the radian. Be able to convert angles from degrees into radians and vice versa.

Outcomes MUST ALL You MUST ALL be able to define the radian AND be able to convert degrees into radians and vice-versa. MOSTSHOULD MOST of you SHOULD Be able to understand the reasons for using radians AND be able to solve problems involving a mixture of degrees and radians. SOMECOULD SOME of you COULD be able to work out arc length.

Radians Radians are units for measuring angles. They can be used instead of degrees. r O 1 radian is the size of the angle formed at the centre of a circle by 2 radii which join the ends of an arc equal in length to the radius. r r x = 1 radian x = 1 rad. or 1 c

r O 2r r 2c2c If the arc is 2r, the angle is 2 radians. Radians

O If the arc is 3r, the angle is 3 radians. r 3r r 3c3c If the arc is 2r, the angle is 2 radians. Radians

O If the arc is 3r, the angle is 3 radians. If the arc is 2r, the angle is 2 radians. r r If the arc is r, the angle is radians. r Radians

O If the arc is 3r, the angle is 3 radians. r r If the arc is 2r, the angle is 2 radians. If the arc is r, the angle is radians. r Radians

If the arc is r, the angle is radians. O r r r But, r is half the circumference of the circle so the angle is Hence, Radians

We sometimes say the angle at the centre is subtended by the arc. Hence, r O r r x x = 1 radian Radians

 Radians SUMMARY One radian is the size of the angle subtended by the arc of a circle equal to the radius 1 radian

Exercises 1. Write down the equivalent number of degrees for the following number of radians: Ans: (a) (b) (c) (d) 2. Write down, as a fraction of, the number of radians equal to the following: (a) (b) (c) (d) Ans: It is very useful to memorize these conversions

Extension Arc Length

Let the arc length be l. O r r l Consider a sector of a circle with angle. Then, whatever fraction is of the total angle at O, l is the same fraction of the circumference. So, ( In the diagram this is about one-third.) circumference

Examples 1. Find the arc length, l, of the sector of a circle of radius 7 cm. and sector angle 2 radians. Solution: where is in radians

2. Find the arc length, l, of the sector of a circle of radius 5 cm. and sector angle. Give exact answers in terms of. Solution: where is in radians rads. So, Examples

 Radians An arc of a circle equal in length to the radius subtends an angle equal to 1 radian. 1 radian  For a sector of angle radians of a circle of radius r, the arc length, l, is given by SUMMARY

1. Find the arc length, l, of the sector shown. O 4 cm l 2. Find the arc length, l, of the sector of a circle of radius 8 cm. and sector angle. Give exact answers in terms of. Exercises

1. Solution: O 4 cm A l Exercises

2. Solution: rads. So, O 8 cm A l where is in radians Exercises