Introduction • This Chapter focuses on using ‘Radians’ when answering questions involving circles • Radians are an alternative to degrees • Radians are quicker to use than degrees (when you get used to them) • They also allow extra calculations which would be much more difficult to do using degrees instead… Radian measure and its Applications You can measure angles in Radians Radians are an alternative to degrees. Some calculations involving circles are easier when Radians are used, as opposed to degrees. r A O 1c r r B ‘If arc AB has length r, then angle AOB is 1 radian (1c or 1 rad)’ Arc Length r 1c Multiply by 2π c Arc Length 2 r 2 360 2 c 180 180 1c c 2πr is the circumference ÷2 ÷π 6A Radian measure and its Applications You can measure angles in Radians Convert the following angle to degrees You need to be able to convert between degrees and radians. Radians Degrees c 1 180 Multiply by 180/π Multiply by 180/ π Top x Top, Bottom x Bottom Cancel out π Work out the sum 7 rad 8 7 180 8 1260 8 1260 8 157.5 6A Radian measure and its Applications You can measure angles in Radians Convert the following angle to degrees You need to be able to convert between degrees and radians. Radians Degrees c 1 180 Multiply by 180/π Multiply by 180/ π Top x Top, Bottom x Bottom Cancel out π Work out the sum 4 rad 15 4 180 15 720 15 720 15 48 6A Radian measure and its Applications You can measure angles in Radians Convert the following angle to radians You need to be able to convert between degrees and radians. Degrees Radians c 1 180 Divide by 180/π Multiply by π/180 Multiply by π/ 180 Only multiply the top here Simplify 150 150 180 150 180 5 rad 6 6A Radian measure and its Applications You can measure angles in Radians Convert the following angle to radians You need to be able to convert between degrees and radians. Degrees Radians c 1 180 Divide by 180/π Multiply by π/180 Multiply by π/ 180 Only multiply the top here Simplify 110 110 180 110 180 11 rad 18 6A Radian measure and its Applications Finding the length of an arc is easier when you use radians Length of Arc Circumference = Angle at Centre Total Angle at Centre = 2 2 r r θ l r l l r = l = r Multiply by 2π Multiply by r (The angle must be in radians!) 6B Radian measure and its Applications Finding the length of an arc is easier when you use radians Find the length of the arc of a circle of radius 5.2cm. The arc subtends an angle of 0.8c at the centre of the circle. l = r l = 5.2 0.8 l = 4.16cm 6B Radian measure and its Applications Finding the length of an arc is easier when you use radians Arc AB of a circle, with centre O and radius r, subtends an angle of θ radians at O. The Perimeter of sector AOB is P cm. Express r in terms of θ. A r O Length AB = rθ Factorise Divide by (θ + 2) P r 2r P r ( 2) P r ( 2) rθ θ r B 6B Radian measure and its Applications Finding the length of an arc is easier when you use radians The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C. C opp hyp 1.2 sin x 2 sin x Inverse sine Double for angle AOB sin x 0.6 Calculator in Radians (H) 2m A x 0.6435rad 1.2m (O) O x B 2 x 1.287rad Angle θ = 2π – 1.287 Angle θ = 4.996 rad 4.996 θ c 2m O 1.287c A 2.4m 2m B (We need to work out angle θ) l r l 2 4.996 l 9.99m 6B Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians Area of Sector Angle at Centre = Total Area Total Angle A O θ Multiply by π X X r2 = 2 X r2 = 2 Multiply by r2 B X = X = r2 2 1 2 r 2 This is the formula’s usual form 6C Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians In the diagram, the area of the minor sector AOB is 28.9cm2. Given that angle AOB is 0.8 rad, calculate the value of r. A Put the numbers in ½ x 0.8 = 0.4 c 0.8 B 28.9 1 2 r 2 1 2 r (0.8) 2 28.9 0.4r 2 Divide by 0.4 r cm O A Square root 72.25 r 2 8.5cm r 6C Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land. The length of the arc must be 66m (adds up to 176 total) Put the numbers in Divide by 55 A 55m 66m 1.2θc O 55m B (We need to work out the angle first) Put the numbers in l r 66 55 1.2c 1 2 A r 2 1 A 552 2 A 1815 m2 6C Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians Area of a Segment Area of Sector AOB – Area of Triangle AOB Area of Sector AOB You can also work out the area of a segment using radians. A 1 2 r 2 Area of Triangle AOB a=b=r C=θ O r A θ r 1 ab sin C 2 1 A r 2 sin 2 A Area of the Segment B Factorise 1 2 1 2 r r sin 2 2 1 2 r ( sin ) 2 6C Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians Calculate the Area of the segment shown in the diagram below. Substitute the numbers in Work the parts out 1 2 r ( sin ) 2 1 2 2.5 sin 2 3 3 3.125 0.1811... Only round the final answer O π 3 2.5cm 0.57 cm2 6C Radian measure and its Applications The Area of a Sector and Segment can be worked out using Radians Area of the shaded segment In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = θ radians. Given that the Area of triangle AOC is 3 three times that of the shaded segment, show that 3θ – 4sinθ = 0. Area of triangle AOC 1 2 r sin 2 a=b=r Angle = π-θ 1 ab sin C 2 1 2 r sin( ) 2 Remember, sin x = sin (180 – x) C A 0 θ 1 2 r sin 2 B AOC = 3 x shaded segment Cancel out 1/2r2 Multiply out the brackets Subtract sinθ 1 2 1 r sin 3 r 2 sin 2 2 sin 3 sin sin 3 3sin 0 3 4sin 6C Summary • We have learnt how to change from degrees to radians • We have seen how to do calculations to work out the length of an arc • We have also seen formulae for the Area or a sector and segment