Core Mathematics 2 Radian Measures Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Radian Measures 1
Radian Measures Radian measure, including use for arc length and area of sector. Use of the formulae s = rθ A = 21 r 2 θ for a circle. Core Mathematics 2 Radian Measures 2
Measuring Angles in Radians A radian is the angle subtended at the centre of a circle by an arc length whose length is equal to that of the radius of the circle. This means a radian is the angle formed when the arc length and the radius are the same. The number of radians in a circle = length of circumference Length of radius = 2 r r = 2 360 = 2 rads 180 = rads 1 rad = 180 57 3 Changing Degrees to Radians Rule:- Multiply by 180 Example 1. Convert 45 to radians 45 180 = 45 180 = 4 Leave your answer in terms of unless asked for more accuracy Example 2. Convert 75 to radians, give your answer to 2sf. 75 180 = 75 180 = 1 308996 = 1 3 (2sf) Rule: Multiply by 180 Example 1. Convert 2 3 Changing Radians to Degrees rads to degress Example 2. 2 180 = 120 3 Covert 2.1 c to degrees 2 1 180 = 120 3 Core Mathematics 2 Radian Measures 3
Finding Arc Lengths The length of an arc is always proportional to the angle at the centre of the arc and the radius of the arc. So, if 2 arcs have the same radius but one has an angle twice the size of the other, it means one arc length will be twice the size of the other. Formula for finding an arc length: l = r where l = arc length, r = radius = angle at the centre Why? l 2 r = 2 ( 2 r) l = 2 r 2 l = r Length of arc = angle at the centre Circumference total angle at centre Ratio of sides = Ratio of angles Example 1. Find the length of the arc ABC. l = r r = 5cm = 2 3 l = 5 2 3 l = 10 3 or 10 47cm Core Mathematics 2 Radian Measures 4
Example 2. Find the radius of the sector ABC l = r r = 16cm = 5 4 16 = r 5 4 16 4 5 = r 64 5 = r or r = 4 07cm Example 3. An arc AB or a circle, with centre o and radius r cm, subtends an angle of θ radians at O. The perimeter of the sector AOB is P cm. Express r in terms of θ. p = (2 x radius) + arc length p = 2r + rθ r = p / (2 + θ) Core Mathematics 2 Radian Measures 5
Finding the Area of a Sector Formula for area of a sector is A = 1 2 r2 where r = radius, = angle at centre Why? A r = 2 2 ( r 2 ) A = 2 r2 A = r2 2 or A = 1 2 r2 Area of Sector = angle at the centre Area of circle total angle at centre Ratio of areas = Ratio of angles NB. If the question gives the angle at the centre in degrees it must be changed to radians Example 1. Find the area of the sector ABC, where ABC = 3 and r = 2cm, give your answer in terms of A = 1 2 r2 where r = 2cm, = 3 A = 1 2 2 2 3 A = 2 3 cm2 Core Mathematics 2 Radian Measures 6
Example 2. Find the area of the sector ABC, where ABC = 60 and r = 8cm, give your answe 60 = 60 180 = 3 A = 1 2 r2 where = 3, r = 8cm A = 1 2 A = 32 3 8 8. A = 33 5 cm 2 (2sf) 3 Core Mathematics 2 Radian Measures 7
Formula for the area of a segment:- Finding the Area of a Segment A = 1 2 r2 1 2 r2 Sin or A = 1 2 r2 ( sin ) where r = radius, = angle at centre Why? Area of segment = Area of sector area of a triangle A = 1 2 r2 1 2 r2 sin as a and b = r ab = r 2 A = 1 2 r2 1 2 r2 sin Example 1. Find the area of the shaded segment. A = 1 2 r2 ( sin ) where r = 9cm and = 6 A = 1 2 9 2 6 sin 6 A = 40 5 (0 023598775) A = 0 95575 (5sf) Core Mathematics 2 Radian Measures 8
Homework Questions 1 Conversion between radians and degrees 1. Change the following angles from degrees into radians. Leave your answers as fractions or multiples of a) 135 b) 315 c) 27 2. Change the following angles from radians into degrees a) 5 4 b) 3 8 c) 3 3. Convert the following angles from degrees into radians. Give your answer correct to 3 sf a) 84 b) 72 c) 405 4. Convert the following angles from radians into degrees a) 3.9 rads b) 1.6 rads c) 5.32 rads Core Mathematics 2 Radian Measures 9
Homework Questions 2 Calculating Arc Lengths 1. Calculate the length of the arcs for the sectors given. Give your answers to 1dp a) r = 6cm, = 5 3 b) r = 13cm, = 2 3 c) r = 7cm, = 3 8 rads 2. Calculate the length of the radius for each of these sectors if:- (give answers to 2dp) a) l = 14mm = 5 7 b) l = 3.8cm = 4 3 c) l = 7.6cm = 5 2 rads 3. Calculate the angle at the centre of the sector if:- (give your answer in degrees to 1dp) a) l = 8cm, r = 5cm b) l = 3.2m, r = 1.8m c) l = 5.1cm, r = 11cm 4. Calculate the radius for each of the following sectors if:- (give answers to 2dp) a) l = 10cm, = 64 b) l = 25cm, = 312 c) l = 13m, = 240 Core Mathematics 2 Radian Measures 10
Homework Questions 3 Finding the Area of a Sector 1. Calculate the area of the sectors, given your answer correct to 3 sf a) r = 4cm = 5 2 b) r = 8cm = 9 4 c) r = 12cm = 8 6 rads 2. Calculate the area of these sectors, give your answers in terms of a) r = 5cm = 72 b) r = 9m = 180 c) r = 7cm = 60 3. Calculate the angle at the centre in degrees, given the area of a sector and the radius a) A = 56 cm 2, r = 9cm b) A = 18 cm 2, r = 12cm c) A = 25.6 cm 2, r = 13cm 4. Calculate angle at the centre in radians, given the area of a sector and the radius a) A = 17 cm 2, r = 3.1cm b) A = 125 cm 2, r = 19cm c) A = 45 m 2, r = 6.9m Core Mathematics 2 Radian Measures 11
Homework Questions 4 Area of a Segment 1. Calculate the area of the segment, given your answer correct to 2 dp a) r = 8m, = 5 4 b) r = 9m, = 11 3 c) r = 112cm, = 17 4 rads d) r = 18.5cm, = 7 12 e) r = 16.2m, = 10 1 rads f) r = 27.6cm, = 13 25 Core Mathematics 2 Radian Measures 12
Past Examination Questions 1. Figure 1 B 8 cm R C 0.7 rad 11 cm D A Figure 1 shows the triangle ABC, with AB = 8 cm, AC = 11 cm and BAC = 0.7 radians. The arc BD, where D lies on AC, is an arc of a circle with centre A and radius 8 cm. The region R, shown shaded in Figure 1, is bounded by the straight lines BC and CD and the arc BD. Find (a) the length of the arc BD, (b) the perimeter of R, giving your answer to 3 significant figures, (c) the area of R, giving your answer to 3 significant figures. (4) (5) Q7,Jan 2005 2. In the triangle ABC, AB = 8 cm, AC = 7 cm, ABC = 0.5 radians and ACB = x radians. (a) Use the sine rule to find the value of sin x, giving your answer to 3 decimal places. (3) Given that there are two possible values of x, (b) find these values of x, giving your answers to 2 decimal places (3) Q7,June 2005 Core Mathematics 2 Radian Measures 13
3. Figure 2 A 6 m B 5 m 5 m O In Figure 2 OAB is a sector of a circle, radius 5 m. The chord AB is 6 m long. (a) Show that cos AO ˆ 7 B =. 25 (b) Hence find the angle (c) Calculate the area of the sector OAB. (d) Hence calculate the shaded area. AO ˆ B in radians, giving your answer to 3 decimal places. (1) (3) Q5, Jan 2006 Core Mathematics 2 Radian Measures 14
4. Figure 2 B C 2.12 m A 1.86 m D Figure 2 shows the cross-section ABCD of a small shed. The straight line AB is vertical and has length 2.12 m. The straight line AD is horizontal and has length 1.86 m. The curve BC is an arc of a circle with centre A, and CD is a straight line. Given that the size of BAC is 0.65 radians, find (a) the length of the arc BC, in m, to 2 decimal places, (b) the area of the sector BAC, in m 2, to 2 decimal places, (c) the size of CAD, in radians, to 2 decimal places, (d) the area of the cross-section ABCD of the shed, in m 2, to 2 decimal places. (3) Q8,May 2006 Core Mathematics 2 Radian Measures 15
5. Figure 2 S P 6 3 m R 6 m 6 m Q Figure 2 shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 6 3 m, (a) find the exact size of angle PQR in radians. (b) Show that the area of the patio PQRS is 12 m 2. (c) Find the exact area of the triangle PQR. (d) Find, in m 2 to 1 decimal place, the area of the segment PRS. (e) Find, in m to 1 decimal place, the perimeter of the patio PQRS. (3) Q9, Jan 2007 Core Mathematics 2 Radian Measures 16
6. C 5 cm 4 cm A B 6 cm Figure 1 Figure 1 shows the triangle ABC, with AB = 6 cm, BC = 4 cm and CA = 5 cm. (a) Show that cos A = 43. (b) Hence, or otherwise, find the exact value of sin A. (3) Q4, May 2007 Core Mathematics 2 Radian Measures 17
7. Figure 1 N C B 500 m 700 m 15 A Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane. Yacht B is 500 m due north of yacht A and yacht C is 700 m from A. The bearing of C from A is 015. (a) Calculate the distance between yacht B and yacht C, in metres to 3 significant figures. (3) The bearing of yacht C from yacht B is, as shown in Figure 1. (b) Calculate the value of. (4) Q6,Jan 2008 Core Mathematics 2 Radian Measures 18
8. Figure 1 Figure 1 shows ABC, a sector of a circle with centre A and radius 7 cm. Given that the size of BAC is exactly 0.8 radians, find (a) the length of the arc BC, (b) the area of the sector ABC. The point D is the mid-point of AC. The region R, shown shaded in Figure 1, is bounded by CD, DB and the arc BC. Find (c) the perimeter of R, giving your answer to 3 significant figures, (d) the area of R, giving your answer to 3 significant figures. (4) (4) Q7,June 2008 Core Mathematics 2 Radian Measures 19
9. Figure 3 The shape BCD shown in Figure 3 is a design for a logo. The straight lines DB and DC are equal in length. The curve BC is an arc of a circle with centre A and radius 6 cm. The size of BAC is 2.2 radians and AD = 4 cm. Find (a) the area of the sector BAC, in cm 2, (b) the size of DAC, in radians to 3 significant figures, (c) the complete area of the logo design, to the nearest cm 2. (4) Q7,Jan 2009 Core Mathematics 2 Radian Measures 20
10. C 4 cm A 0.6 rad 5 cm B 4 cm D Figure 1 An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight line with AB = 5 cm and BD = 4 cm. Angle BAC = 0.6 radians and AC is the longest side of the triangle ABC. 11. (a) Show that angle ABC = 1.76 radians, correct to three significant figures. (b) Find the area of the emblem. (4) (3) Q4, Jan 2010 Figure 1 Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians. (a) Find the length of the arc AB. (b) Find the area of the sector OAB. The line AC shown in Figure 1 is perpendicular to OA, and OBC is a straight line. (c) Find the length of AC, giving your answer to 2 decimal places. The region H is bounded by the arc AB and the lines AC and CB. (d) Find the area of H, giving your answer to 2 decimal places. (3) Q6, Jun 2010 Core Mathematics 2 Radian Measures 21
12. Figure 1 The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle centre O, of radius 6 cm, and angle AOB =. The circle C, inside the sector, touches the two straight edges, 3 OA and OB, and the arc AB as shown. Find (a) the area of the sector OAB, (b) the radius of the circle C. (3) The region outside the circle C and inside the sector OAB is shown shaded in Figure 1. (c) Find the area of the shaded region. Q5, May 2011 Core Mathematics 2 Radian Measures 22
13. Figure 2 Figure 2 shows ABC, a sector of a circle of radius 6 cm with centre A. Given that the size of angle BAC is 0.95 radians, find (a) the length of the arc BC, (b) the area of the sector ABC. The point D lies on the line AC and is such that AD = BD. The region R, shown shaded in Figure 2, is bounded by the lines CD, DB and the arc BC. (c) Show that the length of AD is 5.16 cm to 3 significant figures. Find (d) the perimeter of R, (e) the area of R, giving your answer to 2 significant figures. (4) Q7, Jan 2012 Core Mathematics 2 Radian Measures 23
14. Figure 2 The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY =. The point W lies on the line XY. The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm. (a) Show that, to 3 significant figures, = 2.22 radians. (b) Find the area, in cm 2, of the major sector XZWX. (3) The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shaded in Figure 1. Calculate (c) the area of this shaded region, (d) the perimeter ZWYZ of this shaded region. (3) (4) Q7, Jan 2013 Core Mathematics 2 Radian Measures 24
15. Figure 2 Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12 m and centre B. The points A, B and C lie on a straight line with AB = 23 m and BC = 12 m. Given that the size of angle ABE is exactly 0.64 radians, find (a) the area of the garden, giving your answer in m 2, to 1 decimal place, (b) the perimeter of the garden, giving your answer in metres, to 1 decimal place. (4) (5) Q5,May 2013 Core Mathematics 2 Radian Measures 25
16. Figure 2 Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and BC = 10 m. Given that angle BAC = θ radians, (a) show that, to 3 decimal places, θ = 0.865 (3) The point D lies on AC such that BD is an arc of the circle centre A, radius 7 m. The shaded region S is bounded by the arc BD and the lines BC and DC. The shaded region S will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn, (b) find the amount of grass seed needed, giving your answer to the nearest 10 g. (7) Q8, May 2013_R 17. Core Mathematics 2 Radian Measures 26
Figure 2 The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B. The points A, B and C lie on a straight line with BC = 7.5 cm. Angle EAB = 2 radians, angle EBD = 1.4 radians and CD = 6.1 cm. (a) Find, in cm 2, the area of the sector BDE. (b) Find the size of the angle DBC, giving your answer in radians to 3 decimal places. (c) Find, in cm 2, the area of the shape ABCDEA, giving your answer to 3 significant figures. (5) Q5, May 2014 18. Figure 2 Figure 2 shows the shape ABCDEA which consists of a right-angled triangle BCD joined to a sector ABDEA of a circle with radius 7 cm and centre B. A, B and C lie on a straight line with AB = 7 cm. Given that the size of angle ABD is exactly 2.1 radians, (a) find, in cm, the length of the arc DEA, (b) find, in cm, the perimeter of the shape ABCDEA, giving your answer to 1 decimal place. (4) Q5, May 2014_R Core Mathematics 2 Radian Measures 27
19. Figure 1 Figure 1 shows a sketch of a design for a scraper blade. The blade AOBCDA consists of an isosceles triangle COD joined along its equal sides to sectors OBC and ODA of a circle with centre O and radius 8 cm. Angles AOD and BOC are equal. AOB is a straight line and is parallel to the line DC. DC has length 7 cm. (a) Show that the angle COD is 0.906 radians, correct to 3 significant figures. (b) Find the perimeter of AOBCDA, giving your answer to 3 significant figures. (c) Find the area of AOBCDA, giving your answer to 3 significant figures. (3) (3) Q4, May 2015 Core Mathematics 2 Radian Measures 28