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1Edexcel A Level Specimen Paper 1 Q9
(a) Prove that tan(theta) + cot(theta) = 2cosec(2theta), theta != n*pi/2, n integer. (4) (b) Hence explain why the equation tan(theta) + cot(theta) = 1 does not have any real solutions. (1)
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2EDEXCEL C3 June 2014 Q7
(a) Show that cosec 2x + cot 2x = cot x, x != 90n deg, n integer. (5) (b) Hence, or otherwise, solve, for 0 <= theta < 180deg, cosec(4theta + 10deg) + cot(4theta + 10deg) = sqrt(3). You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
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3EDEXCEL C3 January 2013 Q6
(i) Without using a calculator, find the exact value of (sin 22.5deg + cos 22.5deg)^2. You must show each stage of your working. (5) (ii)(a) Show that cos 2theta + sin theta = 1 may be written in the form k sin^2(theta) - sin theta = 0, stating the value of k. (2) (b) Hence solve, for 0 <= theta < 360deg, the equation cos 2theta + sin theta = 1. (4)
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4EDEXCEL C3 June 2013 Q3
Given that 2cos(x + 50)deg = sin(x + 40)deg, (a) Show, without using a calculator, that tan(x deg) = (1/3)tan 40deg. (4) (b) Hence solve, for 0 <= theta < 360, 2cos(2theta + 50)deg = sin(2theta + 40)deg, giving your answers to 1 decimal place. (4)
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5EDEXCEL C3 January 2012 Q5
Solve, for 0 <= theta <= 180deg, 2cot^2(3theta) = 7cosec(3theta) - 5. Give your answers in degrees to 1 decimal place. (10)
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6EDEXCEL C3 January 2012 Q8
(a) Starting from the formulae for sin(A + B) and cos(A + B), prove that tan(A + B) = (tan A + tan B)/(1 - tan A tan B). (4) (b) Deduce that tan(theta + pi/6) = (1 + sqrt(3)tan(theta))/(sqrt(3) - tan(theta)). (3) (c) Hence, or otherwise, solve, for 0 <= theta <= pi, 1 + sqrt(3)tan(theta) = (sqrt(3) - tan(theta))tan(pi - theta). Give your answers as multiples of pi. (6)
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7EDEXCEL C3 June 2012 Q8
f(x) = 7cos 2x - 24sin 2x. Given that f(x) = R cos(2x + alpha), where R > 0 and 0 < alpha < 90deg, (a) find the value of R and the value of alpha. (3) (b) Hence solve the equation 7cos 2x - 24sin 2x = 12.5 for 0 <= x < 180deg, giving your answers to 1 decimal place. (5) (c) Express 14cos^2(x) - 48sin x cos x in the form a cos 2x + b sin 2x + c, where a, b, and c are constants to be found. (2) (d) Hence, using your answers to parts (a) and (c), deduce the maximum value of 14cos^2(x) - 48sin x cos x. (2)
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8EDEXCEL C3 January 2011 Q1
(a) Express 7cos x - 24sin x in the form R cos(x + alpha) where R > 0 and 0 < alpha < pi/2. Give the value of alpha to 3 decimal places. (3) (b) Hence write down the minimum value of 7cos x - 24sin x. (1) (c) Solve, for 0 <= x < 2pi, the equation 7cos x - 24sin x = 10, giving your answers to 2 decimal places. (5)
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9EDEXCEL C3 January 2011 Q3
Find all the solutions of 2cos(2theta) = 1 - 2sin(theta) in the interval 0 <= theta < 360deg. (6)
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10EDEXCEL C3 June 2011 Q6
(a) Prove that 1/sin(2theta) - cos(2theta)/sin(2theta) = tan(theta), theta != 90n deg, n integer. (4) (b) Hence, or otherwise, (i) show that tan 15deg = 2 - sqrt(3), (3) (ii) solve, for 0 < x < 360deg, cosec 4x - cot 4x = 1. (5)
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11EDEXCEL C3 January 2010 Q3
(a) Express 5cos x - 3sin x in the form R cos(x + alpha), where R > 0 and 0 < alpha < pi/2. (4) (b) Hence, or otherwise, solve the equation 5cos x - 3sin x = 4 for 0 <= x < 2pi, giving your answers to 2 decimal places. (5)
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12EDEXCEL C3 January 2010 Q8
Solve cosec^2(2x) - cot(2x) = 1 for 0 <= x <= 180deg. (7)
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13EDEXCEL C3 January 2009 Q6
(a)(i) By writing 3theta = (2theta + theta), show that sin(3theta) = 3sin(theta) - 4sin^3(theta). (4) (ii) Hence, or otherwise, for 0 < theta < pi/3, solve 8sin^3(theta) - 6sin(theta) + 1 = 0. Give your answers in terms of pi. (5) (b) Using sin(theta - alpha) = sin(theta)cos(alpha) - cos(theta)sin(alpha), or otherwise, show that sin 15deg = (1/4)(sqrt(6) - sqrt(2)). (4)
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14EDEXCEL C3 January 2009 Q8
(a) Express 3cos(theta) + 4sin(theta) in the form R cos(theta - alpha), where R and alpha are constants, R > 0 and 0 < alpha < 90deg. (4) (b) Hence find the maximum value of 3cos(theta) + 4sin(theta) and the smallest positive value of theta for which this maximum occurs. (3) The temperature, f(t), of a warehouse is modelled using the equation f(t) = 10 + 3cos(15t)deg + 4sin(15t)deg, where t is the time in hours from midday and 0 <= t < 24. (c) Calculate the minimum temperature of the warehouse as given by this model. (2) (d) Find the value of t when this minimum temperature occurs. (3)
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15EDEXCEL C3 June 2009 Q2
(a) Use the identity cos^2(theta) + sin^2(theta) = 1 to prove that tan^2(theta) = sec^2(theta) - 1. (2) (b) Solve, for 0 <= theta < 360deg, the equation 2tan^2(theta) + 4sec(theta) + sec^2(theta) = 2. (6)
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16EDEXCEL C3 June 2009 Q6
(a) Use the identity cos(A + B) = cos A cos B - sin A sin B, to show that cos 2A = 1 - 2sin^2(A). (2) The curves C1 and C2 have equations C1: y = 3sin 2x and C2: y = 4sin^2(x) - 2cos 2x. (b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation 4cos 2x + 3sin 2x = 2. (3) (c) Express 4cos 2x + 3sin 2x in the form R cos(2x - alpha), where R > 0 and 0 < alpha < 90deg, giving the value of alpha to 2 decimal places. (3) (d) Hence find, for 0 <= x < 180deg, all the solutions of 4cos 2x + 3sin 2x = 2, giving your answers to 1 decimal place. (4)
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17EDEXCEL C3 June 2009 Q8
(a) Write down sin 2x in terms of sin x and cos x. (1) (b) Find, for 0 < x < pi, all the solutions of the equation cosec x - 8cos x = 0, giving your answers to 2 decimal places. (5)
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18Edexcel AS June 2023 Paper 1 Q3
A sketch shows the position of three phone masts, A, B and C. The masts are identical and their bases are assumed to lie in the same horizontal plane. From mast C: mast A is 8.2 km away on a bearing of 072deg, and mast B is 15.6 km away on a bearing of 039deg. (a) Find the distance between masts A and B, giving your answer in km to one decimal place. (3) An engineer needs to travel from mast A to mast B. (b) Give a reason why the answer to part (a) is unlikely to be an accurate value for the distance the engineer travels. (1)
