C2 Trigonometry: Arc Length & Sector Area PhysicsAndMathsTutor · The diagram above shows a closed...

C2 Trigonometry: Arc Length & Sector Area PhysicsAndMathsTutor.com

Edexcel Internal Review 1

1.

The diagram above 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. (2)

(b) Find the area of the sector OAB. (2)

The line AC shown in the diagram above is perpendicular to OA, and OBC is a straight line.

(c) Find the length of AC, giving your answer to 2 decimal places. (2)

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)

(Total 9 marks)



2.

An emblem, as shown in the diagram above, 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.

(a) Show that angle ABC = 1.76 radians, correct to 3 significant figures. (4)

(b) Find the area of the emblem. (3)

(Total 7 marks)

3.

The diagram above shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height h cm. The cross section is a sector of a circle. The sector has radius r cm and angle 1 radian.

The volume of the box is 300 cm3.



(a) Show that the surface area of the box, S cm2, is given by

rrS 18002 +=

(5)

(b) Use calculus to find the value of r for which S is stationary. (4)

(c) Prove that this value of r gives a minimum value of S. (2)

(d) Find, to the nearest cm2, this minimum value of S. (2)

(Total 13 marks)



4.

The shape BCD shown above 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 cm2, (2)

(b) the size of ∠DAC, in radians to 3 significant figures, (2)

(c) the complete area of the logo design, to the nearest cm2. (4)

(Total 8 marks)



5.

7 cm

B

CD

R

A

0.8 rad

The diagram above 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, (2)

(b) the area of the sector ABC. (2)

The point D is the mid-point of AC. The region R, shown shaded in the diagram above, is bounded by CD, DB and the arc BC.

Find

(c) the perimeter of R, giving your answer to 3 significant figures, (4)

(d) the area of R, giving your answer to 3 significant figures. (4)

(Total 12 marks)



6. A circle C has centre M (6, 4) and radius 3.

(a) Write down the equation of the circle in the form

(x – a)2 + (y – b)2 = r2. (2)

y

3

Q

P(12, 6)

T

C

O

M(6, 4)

x

The diagram above shows the circle C. The point T lies on the circle and the tangent at T passes through the point P (12, 6). The line MP cuts the circle at Q.

(b) Show that the angle TMQ is 1.0766 radians to 4 decimal places. (4)

The shaded region TPQ is bounded by the straight lines TP, QP and the arc TQ, as shown in the diagram above.

(c) Find the area of the shaded region TPQ. Give your answer to 3 decimal places. (5)

(Total 11 marks)



7.

S

Q

P R

6m 6m

6√3m

The diagram above 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√3m,

(a) find the exact size of angle PQR in radians. (3)

(b) Show that the area of the patio PQRS is 12π m2. (2)

(c) Find the exact area of the triangle PQR. (2)

(d) Find, in m2 to 1 decimal place, the area of the segment PRS. (2)

(e) Find, in m to 1 decimal place, the perimeter of the patio PQRS. (2)

(Total 11 marks)



8.

2.12 m

1.86 m

B

C

DA

The figure above 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, (2)

(b) the area of the sector BAC, in m2, to 2 decimal places, (2)

(c) the size of ∠CAD, in radians, to 2 decimal places, (2)

(d) the area of the cross section ABCD of the shed, in m2, to 2 decimal places. (3)

(Total 9 marks)



9.

Figure 1

6 cm1.2

A

B C

6 cm

Figure 1 shows the cross-section ABC of a metal cutter used for making biscuits. The straight sides AB and AC are both of length 6 cm and ∠BAC is 1.2 radians. The curved portion BC is an arc of a circle with centre A.

(a) Find the perimeter of the cross-section of the cutter. (2)

(b) Find the area of the cross-section ABC. (2)

Figure 2

1.2

6 cm

1.2

a cm

b cm

A pair of these cutters are kept together in a rectangular box of length a cm and width b cm. The cutters fit into the box as shown in Figure 2.

(c) Find the value of a and the value of b, giving your answers to 3 significant figures. (4)

(Total 8 marks)



10.

A B

O

6 m

5 m 5 m

In the figure above OAB is a sector of a circle radius 5 m. The chord AB is 6 m long.

(a) Show that cos .257ˆ =BOA

(2)

(b) Hence find the angle BOA ˆ in radians, giving your answer to 3 decimal places. (1)

(c) Calculate the area of the sector OAB. (2)

(d) Hence calculate the shaded area. (3)

(Total 8 marks)



11.

A

D

O C B45m

30m

A fence from a point A to a point B is in the shape of an arc AB of a circle with centre O and radius 45 m, as shown in the diagram. The length of the fence is 63 m.

(a) Show that the size of ∠ AOB is exactly 1.4 radians. (2)

The points C and D are on the lines OB and OA respectively, with OC = OD = 30 m.

A plot of land ABCD, shown shaded in the figure above, is enclosed by the arc AB and the straight lines BC, CD and DA.

(b) Calculate, to the nearest m2, the area of this plot of land. (5)

(Total 7 marks)



12.

L M

O r cm

A major sector LOM of a circle, with centre O and radius r cm, has ∠ LOM = θ radians, as shown in the diagram. The perimeter of the sector is P cm and the area of the sector is A cm2.

(a) Write down, in terms of r and θ, expressions for P and A. (2)

Given that r = 2√2 and that P = A,

(b) show that θ = 12

2−√

.

(3)

(c) Express θ in the form a + b√2, where a and b are integers to be found. (3)

(Total 8 marks)



13.

C

B

D

A

8 cm

0.7 rad

R

11 cm

This diagram 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 the diagram, is bounded by the straight lines BC and CD and the arc BD.

Find

(a) the length of the arc BD, (2)

(b) the perimeter of R, giving your answer to 3 significant figures, (4)

(c) the area of R, giving your answer to 3 significant figures. (5)

(Total 11 marks)



14.

6 cm

2 cm

2 7 cm√

C

A

B

D

In ∆ABC, AB = 2 cm, AC = 6 cm and BC = 2√7 cm.

(a) Use the cosine rule to show that ∠BAC = 3π

radians.

(3)

The circle with centre A and radius 2 cm intersects AC at the point D, as shown in the diagram above.

Calculate

(b) the length, in cm, of the arc BD, (2)

(c) the area, in cm2, of the shaded region BCD. (4)

(Total 9 marks)



15.

O

A B

6.5 cm

The diagram above shows the sector AOB of a circle, with centre O and radius 6.5 cm, and ∠AOB = 0.8 radians.

(a) Calculate, in cm2, the area of the sector AOB. (2)

(b) Show that the length of the chord AB is 5.06 cm, to 3 significant figures. (3)

The segment R, shaded in the diagram above, is enclosed by the arc AB and the straight line AB.

(c) Calculate, in cm, the perimeter of R. (2)

(Total 7 marks)

16.

A BR

O

r r



The diagram above shows the sector OAB of a circle of radius r cm. The area of the sector is 15 cm2 and ∠AOB = 1.5 radians.

(a) Prove that r = 2√5. (3)

(b) Find, in cm, the perimeter of the sector OAB. (2)

The segment R, shaded in the diagram above, is enclosed by the arc AB and the straight line AB.

(c) Calculate, to 3 decimal places, the area of R. (3)

(Total 8 marks)



17.

y

A M D14 14

B C

x

The diagram above shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M.

Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.

(a) Show that the length of BM is 25 mm. (1)

(b) Show that, to 3 significant figures, ∠BMC = 0.568 radians. (3)

(c) Hence calculate, in mm2, the area of the cross-section of the chocolate bar. (5)

Given that this chocolate bar has length 85 mm,

(d) calculate, to the nearest cm3, the volume of the bar. (2)

(Total 11 marks)



18.

A B

C

2a

A flat plate S, which is part of a child’s toy, is shown in the diagram above. The points A, B and C are the vertices of an equilateral triangle and the distance between A and B is 2a. The circular arc AB has centre C and radius 2a. The circular arcs BC and CA have centres at A and B respectively and radii 2a.

(a) Find, in terms of π and a, the perimeter of S. (2)

(b) Prove that the area of the plate S is

2a2(π − 3 ). (6)

(Total 8 marks)



19.

A

B

CD 6 cm

6 cm

6 cm0.4

The diagram above shows a logo ABD.

The logo is formed from triangle ABC. The mid-point of AC is D and BC = AD = DC = 6 cm. ∠BCA = 0.4 radians. The curve BD is an arc of a circle with centre C and radius 6 cm.

(a) Write down the length of the arc BD. (1)

(b) Find the length of AB. (3)

(c) Write down the perimeter of the logo ABD, giving your answer to 3 significant figures. (1)

(Total 5 marks)

C2 Trigonometry: Trigonometric Graphs PhysicsAndMathsTutor.com


1. (a) Sketch, for 0 ≤ x ≤ 2π, the graph of y = sin .6

+

πx

(2)

(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.

(3)

(c) Solve, for 0 ≤ x ≤ 2π, the equation

,65.06

sin =

+

πx

giving your answers in radians to 2 decimal places (5)

(Total 10 marks)

3. The curve C has equation y = cos

+

4πx , 0 ≤ x ≤ 2π.

(a) Sketch C. (2)

(b) Write down the exact coordinates of the points at which C meets the coordinate axes. (3)

(c) Solve, for x in the interval 0 ≤ x ≤ 2π,

cos

+

4πx = 0.5,

giving your answers in terms of p. (4)

(Total 9 marks)

C2 Trigonometry: Trigonometric Equations PhysicsAndMathsTutor.com


7. Find all the solutions, in the interval 0 ≤ x < 2π, of the equation

2 cos2 x + 1 = 5 sin x,

giving each solution in terms of π. (Total 6 marks)

13. Solve, for 0 ≤ θ < 2π, the equation

sin2 θ = 1 + cos θ ,

giving your answers in terms of π. (Total 5 marks)

19. The curve C has equation y = cos

+

4πx , 0 ≤ x ≤ 2π.

(a) Sketch C. (2)

(b) Write down the exact coordinates of the points at which C meets the coordinate axes. (3)

(c) Solve, for x in the interval 0 ≤ x ≤ 2π,

cos

+

4πx = 0.5,

giving your answers in terms of p. (4)

(Total 9 marks)

C2 Trigonometry: Trigonometric Identities PhysicsAndMathsTutor.com


3. Solve, for 0 ≤ θ < 2π, the equation

sin2 θ = 1 + cos θ ,

giving your answers in terms of π. (Total 5 marks)

(ii) Given that sin α =

135

, 0 < α < 2π

, find the exact value of

(a) cos α,

(b) cos 2α. (4)

Given also that 13 cos (x + α) + 5 sin x = 6, and 0 < α < 2π

,

(c) find the value of x. (5)

(Total 14 marks)

C2 Trigonometry: Sine & Cosine Rule PhysicsAndMathsTutor.com


3.

C

5 cm

4 cm A

6 cm B

The diagram above shows the triangle ABC, with AB = 6 cm, BC = 4 cm and CA = 5 cm.

(a) Show that .43cos =A

(3)

(b) Hence, or otherwise, find the exact value of sin A. (2)

(Total 5 marks)

4. 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)

(Total 6 marks)

C2 Trigonometry: Arc Length & Sector Area PhysicsAndMathsTutor · The diagram above shows a closed...

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