Trigonometry Radian Measure Length of Arc Area of Sector Area of Segment
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 , radians in a complete circle. Thus, one radian is about angular degrees.
Radian Measure r r 1 radian
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°
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
r Length of Arc l θ θ must be in radians Fraction of circle Length of arc Circumference = 2πr
r Area of Sector Fraction of circle Area of sector Area of circle = π r 2 θ θ must be in radians
r θ
Examples l = 2·5 8 l = 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.
(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 l = rθ= 10(0·8)= 8 cml = rθ Q1.Q2.
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 = 0·628 radians l = rθ
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 l = rθ l = 9 100 = 900 cm 900 = 2.5 θ θ =θ = 900 2·5 θ = 360 radians
Area of Segment Q5.Find the area of the shaded region. Find the area of the shaded region.