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AS/A Level Mathematics
Sine Rule, Cosine Rule,
Area of any Triangle
Candidates may use any calculator allowed by the regulations of the
Joint Council for Qualifications. Calculators must not have the facility
for symbolic algebra manipulation, differentiation and integration, or
have retrievable mathematical formulae stored in them.
Instructions
• Use black ink or ball-point pen.
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Fill in the boxes at the top of this page with your name.
• Answer all questions and ensure that your answers to parts of questions are
clearly labelled..
• Answer the questions in the spaces provided
– there may be more space than you need.
• You should show sufficient working to make your methods clear.
Answers without working may not gain full credit.
• Answers should be given to three significant figures unless otherwise stated.
Information
• The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
• Read each question carefully before you start to answer it.
• Try to answer every question.
• Check your answers if you have time at the end.
mathsgenie.co.uk
In triangle ABC, side AB has length 15cm, side AC has length 12cm and ∠BAC = 60o
1
(a) Find the length of side BC.
(3)
(b) Find the area of triangle ABC.
(2)
(Total for question 1 is 5 marks)
2
In triangle ABC, side AB has length 8cm, side BC has length 7cm and side AC has length 6cm.
(a) Find the size of angle ABC.
(3)
(b) Find the area of triangle ABC.
(2)
(Total for question 2 is 5 marks)
3
In triangle DEF, ED = 5cm and EF = 6cm.
Given that sin(∠DEF) = 2 and ∠DEF is acute.
3
(a) Find the exact value of cos(∠DEF)
(2)
(b) Find the length of DF.
(4)
(c) Find ∠EFD.
(3)
(Total for question 3 is 9 marks)
4
In triangle PQR, side PQ has length 9cm and side PR has length 10cm.
Given the area of PQR is 30cm2
(a) Find the length of side QR.
(5)
(b) Find ∠PQR
(3)
(Total for question 4 is 8 marks)
5
In the triangle ABC, AB = 13cm, BC =10cm and angle BAC = 30o
Find the two possible sizes of angle ABC, giving your answers to two decimal places.
(Total for question 5 is 6 marks)
6
In the triangle ABC, AB = (x + 3)cm, BC = (x + 2) cm, AC = x cm and angle BAC = 60o
Find the value of x.
(Total for question 6 is 5 marks)
Write your name here
Surname
Other Names
AS/A Level Mathematics
Solving Trigonometric
Equations
Candidates may use any calculator allowed by the regulations of the
Joint Council for Qualifications. Calculators must not have the facility
for symbolic algebra manipulation, differentiation and integration, or
have retrievable mathematical formulae stored in them.
Instructions
• Use black ink or ball-point pen.
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Fill in the boxes at the top of this page with your name.
• Answer all questions and ensure that your answers to parts of questions are
clearly labelled..
• Answer the questions in the spaces provided
– there may be more space than you need.
• You should show sufficient working to make your methods clear.
Answers without working may not gain full credit.
• Answers should be given to three significant figures unless otherwise stated.
Information
• The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
• Read each question carefully before you start to answer it.
• Try to answer every question.
• Check your answers if you have time at the end.
mathsgenie.co.uk
1
Solve, for 0 ≤ x < 180o, the equation,
cos(2x + 15) = 0.3
Give your answers to one decimal place.
(Total for question 1 is 5 marks)
2
Solve, for 0 ≤ θ < 180o, the equation,
sin(3θ – 15) = 0.7
Give your answers to two decimal places.
(Total for question 2 is 5 marks)
3
Solve, for –180 ≤ θ < 180o, the equation,
tan(θ + 30) = –2.5
Give your answers to one decimal place.
(Total for question 3 is 4 marks)
4
Solve, for 0 ≤ x < 360o, the equation,
5cos(x – 40) = 2
Give your answers to two decimal places.
(Total for question 4 is 4 marks)
5
Solve, for 0 ≤ x < 360o, the equation,
tan2(x) = 3
Give your answers to one decimal place.
(Total for question 5 is 5 marks)
6
(a) Show that the equation
2sin2 x = 7cos x + 5
Can be written in the form
2cos2 x + 7cos x + 3 = 0
(3)
(b) Hence solve, for 0 ≤ x < 360o, the equation,
2sin2 x = 7cos x + 5
(5)
(Total for question 6 is 8 marks)
7
(a) Show that the equation
6cos2 x = 4 – sin x
Can be written in the form
6sin2 x – sin x – 2 = 0
(3)
(b) Hence solve, for 0 ≤x < 360o, the equation,
6cos2 x = 4 – sin x
(6)
Give your answers to one decimal place where appropriate.
(Total for question 7 is 9 marks)
8
Find all values for x in the interval 0 ≤ x < 360o, for which
2cos2 x – 3sin2x = 14cos x
Give your answers to one decimal place.
(Total for question 8 is 8 marks)
9
(a) Sketch the graph of y = sin(x – 30) for x in the interval 0 ≤ x < 360o
(2)
(b) Find all values for x in the interval 0 ≤ x < 360o, for which
sin(x – 30) = 0.3
(4)
Give your answers to one decimal place.
(Total for question 9 is 6 marks)
10
Find all values for x in the interval 0 ≤ x < 360o, for which
3tan x = 4sin x
Give your answers to one decimal place where appropriate.
(Total for question 10 is 7 marks)
11
(a) Show that the equation
3sin 2x tan 2 x = cos 2x + 2
Can be written in the form
4cos2 2x + 2cos 2 x – 3 = 0
(4)
o
(b) Find all values for x in the interval 0 ≤ x < 180 , for which
3sin 2x tan 2 x = cos 2x + 2
Give your answers to two decimal places.
(6)
(Total for question 11 is 10 marks)