C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your answers to 1 decimal place. (6) *n3494b00*
physicsandmathstutor.com June 005 5. (a) Using the identity cos(a + B) cosa cosb sin A sinb, prove that cos A 1 sin A. (b) Show that sin θ 3 cos θ 3 sin θ + 3 sin θ (4 cos θ + 6 sin θ 3). 1 (c) Express 4 cos θ + 6 sin θ in the form R sin(θ + α), where R > 0 and 0 < α < π. (d) Hence, for 0 θ < π, solve sin θ = 3(cos θ + sin θ 1), giving your answers in radians to 3 significant figures, where appropriate. (5) 10 *n3494b0100*
physicsandmathstutor.com January 006 6. f(x) = 1 cos x 4 sin x. Given that f(x) = R cos(x + α), where R 0 and 0 α 90, (a) find the value of R and the value of α. (b) Hence solve the equation 1 cos x 4 sin x = 7 for 0 x < 360, giving your answers to one decimal place. (c) (i) Write down the minimum value of 1 cos x 4 sin x. (5) (1) (ii) Find, to decimal places, the smallest positive value of x for which this minimum value occurs. 1 *N3495A010*
physicsandmathstutor.com January 006 7. (a) Show that cos x 1 (i) cos x sin x, x (n 4 )π, n Z, cos x+ sin x 1 (ii) (cos x sin x) cos 1 x cos x sin x. (b) Hence, or otherwise, show that the equation cos θ 1 cosθ = cosθ + sinθ can be written as sin θ = cos θ. (c) Solve, for 0 θ < π, sin θ = cos θ, giving your answers in terms of π. 14 *N3495A0140*
physicsandmathstutor.com June 006 6. (a) Using sin θ + cos θ 1, show that cosec θ cot θ 1. (b) Hence, or otherwise, prove that cosec 4 θ cot 4 θ cosec θ + cot θ. (c) Solve, for 90 < θ < 180, cosec 4 θ cot 4 θ = cot θ. (6) 14 *N3581A0144*
physicsandmathstutor.com June 006 3 8. (a) Given that cos A =, where 70 < A < 360, find the exact value of sin A. 4 π π (b) (i) Show that cos x+ + cos x cos x. 3 3 (5) Given that (ii) show that π π y = x+ x+ + x 3 3 3sin cos cos, dy sin x. dx = *N3581A04*
physicsandmathstutor.com January 007 1. (a) By writing sin 3θ as sin (θ +θ, show that 3 sin3θ = 3sinθ 4sin θ. 3 (b) Given that sin θ =, find the exact value of sin3θ. 4 (5) *N3583A04*
physicsandmathstutor.com January 007 5. Figure 1 Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation y = cos x+ sin x. (a) Express the equation of the curve in the form y = Rsin(x + ), where R and are π constants, R 0and0 < <. (b) Find the values of x, 0 x < π, for which y = 1. 1 *N3583A014*
physicsandmathstutor.com January 007 8. (i) Prove that sec x cosec x tan x cot x. (ii) Given that y = arccos x, 1 x 1 and 0 y π, (a) express arcsin x in terms of y. (b) Hence evaluate arccos x + arcsin x. Give your answer in terms of π. (1) *N3583A04*
physicsandmathstutor.com June 007 6. (a) Express 3 sin x + cos x in the form R sin(x + α) where R > 0 and 0 < α < π. (b) Hence find the greatest value of (3 sin x + cos x) 4. (c) Solve, for 0 < x < π, the equation 3 sin x + cos x = 1, giving your answers to 3 decimal places. (5) 14 *N6109A0144*
physicsandmathstutor.com June 007 7. (a) Prove that sin θ cosθ + = cosec θ, θ 90 n. cosθ sin θ (b) On the axes on page 0, sketch the graph of y = cosec θ for 0 < θ < 360. (c) Solve, for 0 < θ < 360, the equation sinθ cosθ + =3, cosθ sinθ giving your answers to 1 decimal place. (6) 18 *N6109A0184*
physicsandmathstutor.com June 007 Question 7 continued y O 90 180 70 360 θ 0 *N6109A004*
physicsandmathstutor.com January 008 6. (a) Use the double angle formulae and the identity cos( A+ B) cos Acos B sin Asin B to obtain an expression for cos 3x in terms of powers of cos x only. (b) (i) Prove that (ii) Hence find, for 0 cos x 1 sin x + + π sec x, x ( n+ 1). 1 + sin x cos x < x <π, all the solutions of cos x 1 sin x + + = 4. 1+ sin x cos x 14 *H6315RB0144*
. physicsandmathstutor.com June 008 f( x) = 5cosx+ 1sinx Given that f ( x) = Rcos( x α), where R > 0 and 0 α π, (a) find the value of R and the value of α to 3 decimal places. (b) Hence solve the equation for 0 x < π. 5cos x+ 1sin x= 6 (5) (c) (i) Write down the maximum value of 5cos x+ 1sin x. (1) (ii) Find the smallest positive value of x for which this maximum value occurs. 4 *N30745A044*
physicsandmathstutor.com June 008 5. (a) Given that sin θ + cos θ 1, show that 1 + cot θ cosec θ. (b) Solve, for 0 θ < 180, the equation cot θ 9 cosec θ = 3, giving your answers to 1 decimal place. (6) 16 *N30745A0164*
physicsandmathstutor.com January 009 6. (a) (i) By writing 3θ = (θ + θ), show that (ii) Hence, or otherwise, for Give your answers in terms of. sin 3θ = 3 sin θ 4 sin 3 θ. π 0 θ 3, solve 8 sin 3 θ 6 sin θ + 1 = 0. (5) (b) Using sin( θ α) = sin θcosα cosθsin α, or otherwise, show that 1 sin15 =. 4 16 *H3113A0168*
physicsandmathstutor.com January 009 8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ α), where R and α are constants, R > 0 and 0 < α < 90. (b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs. The temperature, f (t), of a warehouse is modelled using the equation f (t) = 10 + 3 cos(15t) + 4 sin(15t), where t is the time in hours from midday and 0 t < 4. (c) Calculate the minimum temperature of the warehouse as given by this model. (d) Find the value of t when this minimum temperature occurs. 4 *H3113A048*
physicsandmathstutor.com June 009 6. (a) Use the identity cos( A+ B) = cos Acos B sin Asin B, to show that cos A = 1 sin A The curves C 1 and C have equations C 1 : y = 3sinx C : y = 4sin x cos x (b) Show that the x-coordinates of the points where C 1 and C intersect satisfy the equation 4cosx + 3sinx = (c) Express 4cosx + 3sinx in the form R cos (x α), where R > 0 and 0 < α < 90, giving the value of α to decimal places. (d) Hence find, for 0 x < 180, all the solutions of 4cosx + 3sinx = giving your answers to 1 decimal place. 18 *H3464A0188*
8. Solve physicsandmathstutor.com January 010 cosec x cot x = 1 for 0 x 180. (7) *N35381A08*
physicsandmathstutor.com June 010 1. (a) Show that sin θ = tanθ 1+ cos θ (b) Hence find, for 180 θ < 180, all the solutions of sin θ = 1 1+ cos θ Give your answers to 1 decimal place. *H35385A08*
physicsandmathstutor.com June 010 π 7. (a) Express sin θ 1.5 cos θ in the form R sin (θ α), where R > 0 and 0 < α <. Give the value of α to 4 decimal places. (b) (i) Find the maximum value of sin θ 1.5 cos θ. (ii) Find the value of θ, for 0 θ <, at which this maximum occurs. Tom models the height of sea water, H metres, on a particular day by the equation 4πt 4πt H = 6+ sin 1. 5cos 5 5, 0 t < 1, where t hours is the number of hours after midday. (c) Calculate the maximum value of H predicted by this model and the value of t, to decimal places, when this maximum occurs. (d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres. (6) *H35385A08*
physicsandmathstutor.com January 011 3. Find all the solutions of cos = 1 sin in the interval 0 360. (6) 8 *H35404RA088*
physicsandmathstutor.com June 011 6. (a) Prove that 1 cos θ = tan θ, θ 90n, n Z sin θ sin θ (b) Hence, or otherwise, (i) show that tan 15 = 3, (ii) solve, for 0 x 360, cosec 4x cot 4x = 1 (5) 1 *P38159A014*
physicsandmathstutor.com January 01 5. Solve, for 0 180, cot 3 7 cosec 3 5 Give your answers in degrees to 1 decimal place. (10) 10 *P40084A0104*
physicsandmathstutor.com January 01 ( )= + ( ) 6. f x x 3x cos 1 x, 0 x (a) Show that the equation f (x) = 0 has a solution in the interval 0.8 < x < 0.9 The curve with equation y = f (x) has a minimum point P. (b) Show that the x-coordinate of P is the solution of the equation x x = 3 + ( 1 sin ) (c) Using the iteration formula x n xn + = + ( 1 3 sin ) 1, x0 = find the values of x 1, x and x 3, giving your answers to 3 decimal places. (d) By choosing a suitable interval, show that the x-coordinate of P is 1.9078 correct to 4 decimal places. 1 *P40084A014*
physicsandmathstutor.com January 01 8. (a) Starting from the formulae for sin ( A + B ) and cos ( A + B ), prove that (b) Deduce that tan A+ tan B tan ( A+ B)= 1 tan Atan B π 1+ 3tanθ tan θ + = 6 3 tanθ (c) Hence, or otherwise, solve, for 0 θ π, tan ( tan ) tan ( ) 1+ 3 θ = 3 θ π θ Give your answers as multiples of (6) 0 *P40084A004*
physicsandmathstutor.com June 01 5. (a) Express 4cosec θ cosec θ in terms of sin and cos. (b) Hence show that 4cosec θ cosec θ = sec θ (c) Hence or otherwise solve, for 0 < <, giving your answers in terms of. 4cosec θ cosec θ = 4 16 *P40686RA0163*
physicsandmathstutor.com June 01 8. f( x) = 7 cosx 4sinx Given that f( x) = Rcos( x+ α ), where R 0 and 0< α < 90, (a) find the value of R and the value of. (b) Hence solve the equation 7 cos x 4sin x= 1. 5 for 0 x 180, giving your answers to 1 decimal place. (c) Express 14cos x 48sin xcos x in the form acos x+ bsin x+ c, where a, b, and c are constants to be found. (5) (d) Hence, using your answers to parts (a) and (c), deduce the maximum value of 14cos x 48sin xcos x 8 *P40686RA083*
physicsandmathstutor.com January 013 4. (a) Express 6 cos + 8 sin in the form R cos( ), where R > 0 and 0 < α < π. Give the value of to 3 decimal places. (b) p( ) = 4, 0 θ π 1 + 6cos + 8sin Calculate (i) the maximum value of p( ), (ii) the value of at which the maximum occurs. 10 *P41486A0108*
physicsandmathstutor.com January 013 6. (i) Without using a calculator, find the exact value of (sin.5 + cos.5 ) You must show each stage of your working. (5) (ii) (a) Show that cos + sin = 1 may be written in the form k sin sin = 0, stating the value of k. (b) Hence solve, for 0 < 360, the equation cos + sin = 1 18 *P41486A0188*
physicsandmathstutor.com June 013 (R)
physicsandmathstutor.com June 013 (R)
physicsandmathstutor.com June 013 (R)
physicsandmathstutor.com June 013 3. Given that (a) Show, without using a calculator, that cos(x + 50) = sin(x + 40) tan x = 1 3 tan (b) Hence solve, for 0 < 360, cos( + 50) = sin( + 40) giving your answers to 1 decimal place. 8 *P43016A083*
physicsandmathstutor.com June 013 8. B 3ms 1 V m s 1 7 m A 4 m Figure Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s 1. Kate is 4 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in Figure. Kate s speed is V ms 1 and she moves in a straight line, which makes an angle, 0 < < 150, with the edge of the road, as shown in Figure. You may assume that V is given by the formula V = 1, 0 < < 150 4sin θ + 7cosθ (a) Express 4sin + 7cos in the form Rcos( ), where R and are constants and where R > 0 and 0 < < 90, giving the value of to decimal places. Given that varies, (b) find the minimum value of V. Given that Kate s speed has the value found in part (b), (c) find the distance AB. Given instead that Kate s speed is 1.68 m s 1, (d) find the two possible values of the angle, given that 0 < < 150. (6) 8 *P43016A083*
Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C. Logarithms and exponentials e x ln a = a x Trigonometric identities sin ( A ± B ) = sin A cos B ± cos A sin B cos( A ± B ) = cos A cos B m sin A sin B tan A ± tan B tan ( A ± B ) = ( A ± B ( k + ) 1m tan A tan B A + B A B sin A + sin B = sin cos A + B A B sin A sin B = cos sin A + B A B cos A + cos B = cos cos A + B A B cos A cos B = sin sin 1 π ) Differentiation f(x) tan kx sec x cot x cosec x f( x) g( x) f (x) k sec kx sec x tan x cosec x cosec x cot x f ( x )g( x ) f( x )g ( x ) (g( x)) 6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 009