1. Trigonometry Radian Measure Length of Arc Area of Sector

2. Radian Measure  To talk about trigonometric functions, it is helpful to move to a different system of angle measure, called radian measure.  A radian is the measure of a central angle whose minor arc is equal in length to the radius of the circle.  There are 2  or approximately 6.28318, radians in a complete circle. Thus, one radian is about 57.296 angular degrees.

3. Radian Measure r r 1 radian

4. Radian Measure  There are 2π radians in a full rotation – once around the circle  There are 360° in a full rotation  To convert from degrees to radians or radians to degrees, use the proportion radians degrees 180    2π = 360°π = 180°

5. Examples  Find the radian measure equivalent of 210°.  Find the degree measure equivalent of radians. 3π 4  ° 4 3  180  4 3π3π 180° = π π 180 °° 210π 180  °  7π 6 

6. r Length of Arc l θ θ must be in radians Fraction of circle Length of arc Circumference = 2πr

7. r Area of Sector Fraction of circle Area of sector Area of circle = π r 2 θ θ must be in radians

8. r θ

9. Examples s = 2·5  8 s = rθ = 20 cm l 2·5 8 cm  A circle has radius length 8 cm. An angle of 2.5 radians is subtended by an arc. Find the length of the arc.

10. (i)Find the length of the minor arc pq. (ii)Find the area of the minor sector opq. p qo 10 cm 0·8 rad p qo 12 cm s = rθ= 10(0·8)= 8 cms = rθ Q1.Q2.

11. Q3. The bend on a running track is a semi-circle of radius A runner, on the track, runs a distance of 20 metres on the bend. The angles through which the runner has run is A. Find to three significant figures, the measure of A in radians. 20 mA 100 π metres. 20 = θ 100 π π 100 θ = 20  = 0·6283.. = 0·628 radians s = rθ

12. 2·5 9 Q4.A bicycle chain passes around two circular cogged wheels. Their radii are 9 cm and 2·5 cm. If the larger wheel turns through 100 radians, through how many radians will the smaller one turn? 100 radians s = rθ s = 9  100 = 900 cm 900 = 2·5θ θ =θ = 900 2·5 θ = 360 radians

13. The diagram shows a sector (solid line) circumscribed by a circle (dashed line). kk 60º r r 30º k2k2 r k2k2 3 2  3 2 cos 30º  krkr 3 1  3 k r r  (i) Find the radius of the circle in terms of k. k2k2 r cos 30º 

14. The diagram shows a sector (solid line) circumscribed by a circle (dashed line). kk r r (ii) Show that the circle encloses an area which is double that of the sector. Area of circle  π r 2 3 k r r      π 2 3 k Area of sector Twice area of sector 