weebly.com/ Core Mathematics 3 Trigonometry

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1 weebly.com/ Core Mathematics 3 Trigonometry Core Maths 3 Trigonometry Page 1

2 C3 Trigonometry In C you were introduced to radian measure and had to find areas of sectors and segments. In addition to this you solved trigonometric equations using the identities below. Sin Cos 1 Sin Tan Cos By the end of this unit you should: Have a knowledge of secant, cosecant and cotangent and of arcsin, arcos and arctan. Their relationship to sine, cosine and tangent and their respective graphs including appropriate restrictions of the domain. Have a knowledge of 1 Cot Co sec and Tan 1 Sec. Have a knowledge of double angle formulae and r formulae. Core Maths 3 Trigonometry Page

3 New trigonometric functions The following three trigonometric functions are the reciprocals of sine, cosine and tan. The way to remember them is by looking at the third letter. Secx 1 Cosx Co sec x 1 Sinx Cotx 1 Tanx Core Maths 3 Trigonometry Page 3

4 These three trigonometric functions are use to derive two more identities. New Identities Starting with Sin Cos 1 and by dividing by Sin gives: Sin Cos Sin Sin 1 Sin Using the new functions outlined above and the fact that this becomes: 1 Cot Co sec. Cos Cot Sin Returning to Sin Cos 1 and by dividing by Cos gives: Therefore: Sin Cos Cos Cos 1 Cos Tan 1 Sec. The three identities will be used time and again. Try to remember them but you should also be able to derive them as outlined above. Sin Cos 1 1 Cot Co sec Tan 1 Sec Example 1 Solve for the equation 5tan sec 1, giving your answers to 1 decimal place. You should have come across questions of this type in C using the identity cos sin 1. The given equation has a single power of secθ therefore we must use an identity to get rid of the tan θ. tan 1 sec So the equation becomes: Core Maths 3 Trigonometry Page 4

5 5sec 5 sec 1 5sec sec 6 0 We now have a quadratic in sec so by factorising: 5sec sec 6 0 (5 sec 6)(sec 1) sec = cos = 146.4, sec 1 cos =1 0,360 Inverse Trigonometric Functions. Functions are introduced in C3 and we use the concept of inverses to find the following functions (remember that the inverse of a function in graphical terms is its reflection in the line y = x). The domain of the original trigonometric function has to be restricted to ensure that it is still one to one. It is also worth remembering that the domain and range swap over as you go from the function to the inverse. ie in the first case π π the domain of sinx is restricted to sin x and this becomes the range of the inverse function. y=arcsinx Domain 1 x 1 Range π arcsin x π Core Maths 3 Trigonometry Page 5

6 Example Find arcsin0.5 = y Simply swap around Siny = 0.5 y = П/6 y=arccosx Domain 1 x 1 Range 0 arccos x π y=arctanx Domain Range x arctan x Core Maths 3 Trigonometry Page 6

7 Addition Formulae A majority of the formulae in C3 need to be learnt. One s in red are in the formula book. Sin( A B) SinACosB CosASinB Cos ( A B) CosACosB SinASinB TanA TanB Tan ( A B) TanATanB The examples below use addition formulae. Example 1 4 Given that Sin A = and that Cos B = where A is obtuse and B is 13 5 reflex find: a) Sin (A + B) b) Cos (A B) c) Cot (A B) Before we start the question it is advisable to draw the graphs of Sin x and Cos x. Core Maths 3 Trigonometry Page 7

8 Since A is obtuse the cosine of A must be negative and by using the 5 Pythagorean triple Cos A =. The angle B is slightly more tricky. We 13 are told that B is reflex but we know that Cos B is positive. Therefore B must be between 70 and 360 and so Sin B is negative. 3 Hence Sin B =. 5 We are now ready to attempt part (a) 1 5 Sin A = Cos A = Sin B = 5 Cos B = 5 4 a) Using the formula above to find Sin (A + B) Sin (A + B) = Sin A Cos B + Sin B Cos A Sin (A + B) = Sin (A + B) = 65 b) Cos (A B) = Cos A Cos B + Sin A Sin B Cos (A B) = Cos (A B) = 65 1 c) Cot (A B) = Tan ( A B) 1 TanATanB TanA TanB Core Maths 3 Trigonometry Page 8

9 The graphs above show the range of values that A and B lie within so we need to consider the Tan graph at these points. I have included the diagrams below to help in my calculations A 5 B 3 Therefore 1 Tan A = since for obtuse angles the tan graph is 5 negative and: 4 Tan B = for the same reason. 3 So finally Cot (A B) = Cot (A B) = Cot (A B) = 1 TanATanB TanA TanB Core Maths 3 Trigonometry Page 9

10 The above example may appear to be a little mean by being non calculator but it is an opportunity to really start thinking about the angles and graphs involved. Double angle Formulae The Addition Formulae are used to derive the double angle formulae. In all cases let A=B, therefore: SinA SinACosA CosA Cos A Sin A TanA TanA 1Tan A The second of the identities above is combined with Sin Cos 1 to express Sin θ and Cos θ in terms of Cosθ. This is vital when you are asked to integrate Sin θ and Cos θ. If we start by trying to express the right hand side of the identity Cos Cos Sin in terms of Cos θ only. Cos Cos Sin Cos Cos 1 - Sin Cos Rearranging to make Cos θ the subject: Cos Cos A similar approach is used to find the identity for Sin θ: Sin Cos Please note that this derivation has been tested on a C3 past paper. Core Maths 3 Trigonometry Page 10

11 Example 3 Given that sin x =, use an appropriate double angle formula to find the 5 exact value of sec x. 1 From earlier work you should remember that Secx and hence Cosx 1 Secx. Cosx Using the identity above: Cos Cos Sin and the fact that: Cos 1- Sin Therefore: Cos 1 Sin and 1 Secx 1 Sin Here s one to complete yourselves! Prove that cot x + cosec x cot x, n (x, n ). Core Maths 3 Trigonometry Page 11

12 The following examples deal with a variety of the identities outlined above. Be warned, some are more complex than others. Example Given that tan x = ½, show that tan x = 5 Using the double angle formula: tanx tanx 1 tan x 1 tan x 1 tan x 1 tan x 4tan x tan x 4 tan x 1 0 Using the quadratic formula to solve a quadratic in tan: tan x 4 tan x 1 0 tan x tan x tan x 5 Example 3 (i) Given that cos(x + 30)º = 3 cos(x 30)º, prove that tan x º = - 3. Using double angle formulae: cosx cos30 sinx sin30 = 3cosxcos30 + 3sinxsin30 cosx cos30 = -4 sinx sin30 sin 30 = ½, cos 30 = 3 Core Maths 3 Trigonometry Page 1

13 3 cosx = -sinx tan x º = cos (ii) (a) Prove that = tan θ. sin Using the fact that Rewriting the fraction: cos 1 sin 1 cos 1 sin 1 sin sin cos sin tan cos (b) Verify that θ = 180º is a solution of the equation sin θ = cos θ. Too easy! Simply let = 180º (c) Using the result in part (a), or otherwise, find the other two solutions, 0 < θ < 360º, of the equation sin θ = cos θ. sin θ = cos θ sin θ = (1 cos θ) 1 1 cos sin Therefore from part (a): tanθ = 0.5 θ = 6.6º, 06.6º Core Maths 3 Trigonometry Page 13

14 Example 4 Find the values of tan θ such that sin θ - sinθsecθ = sinθ -. Remembering that equation becomes: 1 sec cos and the double angle formulae, the sin tan 4 sin cos Dividing by cos tan tan sec 4 tan sec sec 1 tan 3 tan tan tan 4 tan tan 3 tan 4tan 5tan 0 We now have a trigonometric polynomial: 3 t 4t 5t 0 By using factor theorem (t 1) is a factor, therefore: Hence (t 1)(t 3t + ) = (t 1)(t 1)(t ) tan θ = 1 tan θ = Example 5 3 (i) Given that sin x =, use an appropriate double angle formula to 5 find the exact value of sec x. We can use a 3,4,5 Pythagorean triangle to show that cos x = 4 5 sec x = 1 cosx Core Maths 3 Trigonometry Page 14

15 1 1 cosx cos x sin x (ii) Prove that cot x + cosec x cot x, n (x, n ). Left hand side becomes: cosx 1 sinx sinx using cosx cos x 1 and common denomenator of sinx cos x cos x cosx cotx sinx sin xcos x sin x Core Maths 3 Trigonometry Page 15

16 R Formulae Expressions of the type asin bcos can be written in terms of sine or cosine only and hence equations of the type asin bcos c can be solved. The addition formulae outlined above are used in the derivation. In most you cases you will be told which addition formula to use. Example 6 f(x) = 14cosθ 5sinθ Given that f(x) = Rcos(θ + α), where R 0 and 0 90, a) find the value of R and α. b) Hence solve the equation 14cosθ 5sinθ = 8 for 0 360, giving your answers to 1 dp. c) Write down the minimum value of 14cosθ 5sinθ. d) Find, to dp, the smallest value positive value of θ for which this minimum occurs. a) Using the addition formulae: Rcos(θ + α) = R(cosθcosα sinθsinα) Therefore since Rcos(θ + α) = f(x) R(cosθcosα sinθsinα) = 14cosθ 5sinθ Hence Rcosα = 14 Rsinα = 5 Dividing the two tanα = 5/14 α = 19.7º Using Pythagoras R = ( ) = 14.9 Therefore f(x) = 14.9cos(θ ) b) Hence solve the equation 14cosθ 5sinθ = 8 Therefore: 14.9cos(θ ) = 8 cos(θ ) = Core Maths 3 Trigonometry Page 16

17 θ = 37.9º The cos graph has been translated 19.7º to the left, the second solution is at 8.7º (can you see why?) Example 7 a) Express 3 sinx + 7 cosx in the form R sin(x + α), where R > 0 and 0 < α <. Give the values of R and α to 3 dp. b) Express 6 sinxcosx + 14 cos x in the form a cosx + b sinx +c, where a, b and c are constants to be found. c) Hence, using your answer to (a), deduce the maximum value of 6sinxcosx + 14cos x. a) R sin(x + α) = R(sinxcosα + sinαcosx). Hence R(sinxcosα + sinαcosx) = 3 sinx + 7 cosx Therefore: 3 = Rcosα. Because the sinx is being multiplied by the and 7 = Rsinα Dividing the two tanα = 7 3 α = 1.17 c By Pythagoras R = (7 + 3 ) = 7.6 Therfore: 3 sinx + 7 cosx = 7.6 sin(x ) Core Maths 3 Trigonometry Page 17

18 b) Express 6 sinxcosx + 14 cos x in the form a cosx + b sinx +c The question is using part (a) but you have to remember your identities: 6 sinxcosx = 3 sinx cos 1 x = cosx + 1 Therefore: 14cos x = 7cosx sinxcosx + 14 cos x = 3 cosx + 7 sinx + 7 c) Hence, using your answer to (a), deduce the maximum value of 6sinxcosx + 14cos x. From (a) 3 sinx + 7 cosx = 7.6 sin(x ) Therefore: 6 sinxcosx + 14 cos x = 7.6 sin(x ) + 7 Using the right hand side this is a sine curve of amplitude 7.6, it has also been translated 7 units up. Therefore its maximum value will be Questions of the type Rcos(x ) or Rsin(x ) are definitely going to be on a C3 paper. Other trig questions require a little bit of proof and the use of identities. Core Maths 3 Trigonometry Page 18

19 Example 8 Solve, for 0 < θ < π, sin θ + cos θ + 1 = 6 cos θ, giving your answers in terms of π. Since there is a single power of cos θ I will aim to write as much of the equation in cos θ cosx cos x 1 Factorising gives: Therefore: sin θcos θ + cos θ = 6 cos θ Cos θ(sin θ + cos θ - 6) =0 3 cos θ = 0 θ =, or sin θ + cos θ - 6 = 0 sin θ + cos θ = 6 Use of Rsin(θ + α) R = α = 4 sin(θ + 4 ) = 6 5 θ =, 1 1 This is a little challenging but I m sure that parts of it are accessible. Core Maths 3 Trigonometry Page 19

20 Edexcel past examination questions section_01 1. (a) Given that sin + cos 1, show that 1 + tan sec. (b) Solve, for 0 < 360, the equation () tan + sec = 1, giving your answers to 1 decimal place. (6) [005 June Q1]. (a) Show that (i) cos x cos x sin x, x (n cos x sin x 41 ), n Z, (ii) 1 (cos x sin x) cos x cos x sin x 1. (b) Hence, or otherwise, show that the equation () cos 1 cos cos sin can be written as (c) Solve, for 0 <, sin = cos. sin = cos, giving your answers in terms of. [006 January Q7] Core Maths 3 Trigonometry Page 0

21 3. (a) Using sin + cos 1, show that the cosec cot 1. (b) Hence, or otherwise, prove that cosec 4 cot 4 cosec + cot. (c) Solve, for 90 < < 180, cosec 4 cot 4 = cot. () () (6) [006 June Q6] 4. (a) Given that cos A = 43, where 70 < A < 360, find the exact value of sin A. (b) (i) Show that cos x + cos x cos x. 3 3 Given that (ii) show that y = 3 sin x + cos d y = sin x. dx x + cos x, 3 3 [006 June Q8] 5. (a) By writing sin 3 as sin ( + ), show that (b) Given that sin = sin 3 = 3 sin 4 sin 3. 3, find the exact value of sin 3. 4 () [007 January Q1] Core Maths 3 Trigonometry Page 1

22 6. (a) Prove that sin + cos cos = cosec, 90n. sin (b) Sketch the graph of y = cosec θ for 0 < θ < 360. () (c) Solve, for 0 < θ < 360, the equation sin + cos cos = 3 sin giving your answers to 1 decimal place. (6) [007 June Q7] 7. (a) Use the double angle formulae and the identity cos(a + B) cosa cosb sina sinb to obtain an expression for cos 3x in terms of powers of cos x only. (b) (i) Prove that cos x 1 sin x + 1 sin x cos x sec x, x (n + 1). (ii) Hence find, for 0 < x < π, all the solutions of cos x 1 sin x + 1 sin x cos x = 4. [008 January Q6] 8. (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) [008 June Q5] Core Maths 3 Trigonometry Page

23 9. (a) (i) By writing 3θ = (θ + θ), show that sin 3θ = 3 sin θ 4 sin 3 θ. (ii) Hence, or otherwise, for 0 < θ < 3, solve 8 sin 3 θ 6 sin θ + 1 = 0. Give your answers in terms of π. (b) Using sin (θ ) = sin θ cos cos θ sin, or otherwise, show that sin 15 = 4 1 (6 ). [009 January Q6] 10. (a) Use the identity cos θ + sin θ = 1 to prove that tan θ = sec θ 1. (b) Solve, for 0 θ < 360, the equation () tan θ + 4 sec θ + sec θ =. (6) [009 June Q] 11. (a) Write down sin x in terms of sin x and cos x. (b) Find, for 0 < x < π, all the solutions of the equation (1) cosec x 8 cos x = 0. giving your answers to decimal places. [009 June Q8] 1. Solve for 0 x 180. cosec x cot x = 1 (7) [010 January Q8] Core Maths 3 Trigonometry Page 3

24 13. (a) Show that sin θ = tan θ. 1 cos θ (b) Hence find, for 180 θ < 180, all the solutions of () sin θ 1 cos θ = 1. Give your answers to 1 decimal place. [010 June Q1] 14. Find all the solutions of cos = 1 sin in the interval 0 < (a) Prove that (b) Hence, or otherwise, 1 cos = tan, 90n, n Z. sin sin (i) show that tan 15 = 3, (ii) solve, for 0 < x < 360, (6) [011January Q3] 16. Solve, for 0 180, cosec 4x cot 4x = 1. cot 3 = 7 cosec 3 5. [011 June Q6] Give your answers in degrees to 1 decimal place. (10) [01January Q5] Core Maths 3 Trigonometry Page 4

25 17. (a) Starting from the formulae for sin (A + B) and cos (A + B), prove that (b) Deduce that tan (A + B) = tan A tan B. 1 tan Atan B 1 3tan tan =. 6 3 tan (c) Hence, or otherwise, solve, for 0 θ π, tan θ = ( 3 tan θ) tan (π θ). Give your answers as multiples of π. (6) [01January Q8] 18. (a) Express 4 cosec θ cosec θ in terms of sin θ and cos θ. (b) Hence show that 4 cosec θ cosec θ = sec θ. (c) Hence or otherwise solve, for 0 < θ <, () 4 cosec θ cosec θ = 4 giving your answers in terms of. [01June Q5] Core Maths 3 Trigonometry Page 5

26 19. (i) Without using a calculator, find the exact value of (sin.5 + cos.5 ). You must show each stage of your working. (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. () [013January Q6] 0. Given that cos(x + 50) = sin(x + 40) (a) Show, without using a calculator, that tan x = 1 3 tan 40 (b) Hence solve, for 0 θ < 360, cos(θ + 50) = sin(θ + 40) giving your answers to 1 decimal place. [013June Q3] 1. (i) Use an appropriate double angle formula to show that cosec x = λ cosec x sec x, and state the value of the constant λ. (ii) Solve, for 0 θ < π, the equation 3sec θ + 3 sec θ = tan θ You must show all your working. Give your answers in terms of π. (6) [013_R June Q6] Core Maths 3 Trigonometry Page 6

27 . (a) Show that cosec x + cot x = cot x, x 90n, n (b) Hence, or otherwise, solve, for 0 θ < 180, cosec (4θ + 10 ) + cot (4θ + 10 ) = 3 You must show your working. [014June Q7] 3. (i) (a) Show that tan x cot x = 5 cosec x may be written in the form a cos x + b cos x + c = 0 stating the values of the constants a, b and c. (b) Hence solve, for 0 x < π, the equation tan x cot x = 5 cosec x (ii) Show that 4. Given that giving your answers to 3 significant figures. tan θ + cot θ λ cosec θ, stating the value of the constant λ. n, n tan = p, where p is a constant, p 1, [014_R June Q3] use standard trigonometric identities, to find in terms of p, (a) tan, (b) cos, (c) cot ( 45). Write each answer in its simplest form. () () () [015June Q1] Core Maths 3 Trigonometry Page 7

28 5. (a) Prove that sec A + tan A cos A sin A, A cos A sin A ( n 1), n Z. 4 (b) Hence solve, for 0 <, sec + tan = 1. Give your answers to 3 decimal places. [015, June Q8] 6. (a) Prove that cot x + tan x cot x, x n, n Z. (b) Hence, or otherwise, solve, for π x < π, Give your answers to 3 decimal places. 6 cot x + 3 tan x = cosec x. (Solutions based entirely on graphical or numerical methods are not acceptable.) (6) 7. (a) Prove that [016, June Q8] sin x tan x º tan x cos x, x ¹ (n + 1)90, n Z (b) Given that x ¹ 90 and x ¹ 70, solve, for 0 x < 360, sin x tan x = 3 tan x sin x Give your answers in degrees to one decimal place where appropriate. (Solutions based entirely on graphical or numerical methods are not acceptable.) [017, June Q9] 8. Solve, for 0 cos 5 13sin Give your answers in radians to 3 decimal places. [015, Jan. IAL Q] Core Maths 3 Trigonometry Page 8

29 9. (a) Given that cos (x + 30) = sin (x 30) without using a calculator, show that tan x = 33 4 (b) Hence or otherwise solve, for 0 θ < 180, cos (θ + 40) = sin (θ 0) Give your answers to one decimal place. 30. (a) Use the substitution t = tan x to show that the equation [015, Jan, IAL Q7] can be written in the form 4tan x 3cot x sec x = 0 (b) Hence solve, for 0 x < π, 3t 4 + 8t 3 = 0 4tan x 3cot x sec x = 0 Give each answer in terms of π. You must make your method clear. [015, June, IAL Q7] 31. (a) Show that cot x cosec x 11 = 0 may be expressed in the form cosec x cosec x + k = 0, where k is a constant. (b) Hence solve for 0 x < 360 (1) cot x cosec x 11 = 0 Give each solution in degrees to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [016, Jan, IAL Q] Core Maths 3 Trigonometry Page 9

30 3. (a) Prove that sin x tan x tan x cos x, (b) Hence solve, for 0 θ < (n 1) x, n Z (i) sin θ tan θ = 3 cos θ (ii) tan(θ + 1) cos(θ + ) sin(θ + ) = Give your answers in radians to 3 significant figures, as appropriate. (Solutions based entirely on graphical or numerical methods are not acceptable) 33. (a) Using the trigonometric identity for tan(a + B), prove that tan3x = 3tan x - tan3 x, x ¹ (n +1)30, 1-3tan x (b) Hence solve, for 30 < x < 30, tan 3x = 11 tan x 34. (a) Prove that (7) [016, June, IAL Q8] (Solutions based entirely on graphical or numerical methods are not acceptable.) 1- cosx 1+ cosx º tan x, x (n + 1)90, n Z (b) Hence, or otherwise, solve, for 90 < θ < 90, - cosq 1+ cosq - = 7secq Give your answers in degrees to one decimal place. [017, Jan, IAL Q8] (Solutions based entirely on graphical or numerical methods are not acceptable.) (6) [017, June, IAL Q7] Core Maths 3 Trigonometry Page 30

31 Edexcel past examination questions section_0 1. (a) Using the identity cos (A + B) cos A cos B sin A sin B, prove that (b) Show that cos A 1 sin A. () sin 3 cos 3 sin + 3 sin (4 cos + 6 sin 3). (c) Express 4 cos + 6 sin in the form R sin ( + ), where R > 0 and 0 < < 1. (d) Hence, for 0 <, solve sin = 3(cos + sin 1), giving your answers in radians to 3 significant figures, where appropriate.. f(x) = 1 cos x 4 sin x. Given that f(x) = R cos (x + ), where R 0 and 0 90, [005 June Q5] (a) find the value of R and the value of. (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. (ii) Find, to decimal places, the smallest positive value of x for which this minimum value occurs. () [006January Q6] (1) Core Maths 3 Trigonometry Page 31

32 3. Figure 1 y x The curve on the screen satisfies the equation y = 3 cos x + sin x. (a) Express the equation of the curve in the form y = R sin (x + ), where R and are constants, R > 0 and 0 < <. (b) Find the values of x, 0 x <, for which y = 1. [007January Q5] 4. (a) Express 3 sin x + cos x in the form R sin (x + α) where R > 0 and 0 < α <. (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. () [007 June Q7] Core Maths 3 Trigonometry Page 3

33 5. A curve C has equation The point A(0, 4) lies on C. y = 3 sin x + 4 cos x, π x π. (a) Find an equation of the normal to the curve C at A. (b) Express y in the form R sin(x + α), where R > 0 and 0 < <. Give the value of to 3 significant figures. (c) Find the coordinates of the points of intersection of the curve C with the x-axis. Give your answers to decimal places. 6. f(x) = 5 cos x + 1 sin x. [008January Q7] Given that f(x) = R cos (x α), where R > 0 and 0 < α <, (a) find the value of R and the value of α to 3 decimal places. (b) Hence solve the equation for 0 x < π. 5 cos x + 1 sin x = 6 (c) (i) Write down the maximum value of 5 cos x + 1 sin x. (ii) Find the smallest positive value of x for which this maximum value occurs. () (1) [008 June Q] 7. (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 Core Maths 3 Trigonometry Page 33

34 f (t) = 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. () 8. (a) Use the identity cos (A + B) = cos A cos B sin A sin B, to show that [009January Q8] cos A = 1 sin A () The curves C1 and C have equations C1: y = 3 sin x C: y = 4 sin x cos x (b) Show that the x-coordinates of the points where C1 and C intersect satisfy the equation 4 cos x + 3 sin x = (c) Express 4cos x + 3 sin x 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 4 cos x + 3 sin x =, giving your answers to 1 decimal place. [009 June Q6] 9. (a) Express 5 cos x 3 sin x in the form R cos(x + α), where R > 0 and 0 < α < 1. (b) Hence, or otherwise, solve the equation 5 cos x 3 sin x = 4 for 0 x <, giving your answers to decimal places. [010January Q3] Core Maths 3 Trigonometry Page 34

35 10. (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 < π, at which this maximum occurs. Tom models the height of sea water, H metres, on a particular day by the equation H = 6 + sin 4 t 5 4 t 1.5 cos, 0 t <1, 5 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) [010 June Q7] 11. (a) Express 7 cos x 4 sin x in the form R cos (x + ) where R > 0 and 0 < <. Give the value of to 3 decimal places. (b) Hence write down the minimum value of 7 cos x 4 sin x. (c) Solve, for 0 x <, the equation (1) 7 cos x 4 sin x = 10, giving your answers to decimal places. [011January Q1] Core Maths 3 Trigonometry Page 35

36 1. (a) Express cos 3x 3 sin 3x in the form R cos (3x + ), where R and are constants, R > 0 and 0 < <. Give your answers to 3 significant figures. f(x) = e x cos 3x. (b) Show that f (x) can be written in the form f (x) = Re x cos (3x + ), where R and are the constants found in part (a). (c) Hence, or otherwise, find the smallest positive value of x for which the curve with equation y = f(x) has a turning point. [011 June Q8] 13. f(x) = 7 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 7 cos x 4 sin x = 1.5 for 0 x < 180, giving your answers to 1 decimal place. (c) Express 14 cos x 48 sin x cos x in the form a cos x + b sin x + c, where a, b, and c are constants to be found. () (d) Hence, using your answers to parts (a) and (c), deduce the maximum value of 14 cos x 48 sin x cos x. () [01 June Q8] Core Maths 3 Trigonometry Page 36

37 14. (a) Express 6 cos + 8 sin in the form R cos ( α), where R > 0 and 0 < α <. Give the value of α to 3 decimal places. (b) 4 p( ) =, 1 6cos 8sin 0. Calculate (i) the maximum value of p( ), (ii) the value of at which the maximum occurs. [013January Q4] 15. 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 m s 1 and she moves in a straight line, which makes an angle θ, 0 < θ < 150, with the edge of the road, as shown in Figure. You may assume that V is given by the formula V 1, 0 < θ < 150 4sin 7cos (a) Express 4sin θ + 7cos θ in the form Rcos(θ α), where R and α are constants and where R > 0 and 0 < α < 90, giving the value of α to decimal places. Core Maths 3 Trigonometry Page 37

38 Given that θ 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 θ, given that 0 < θ < 150. (6) [013 June Q8] 16. f(x) = 7cos x + sin x Given that f(x) = Rcos(x a), where R > 0 and 0 < a < 90, (a) find the exact value of R and the value of a to one decimal place. (b) Hence solve the equation 7cos x + sin x = 5 for 0 x < 360, giving your answers to one decimal place. (c) State the values of k for which the equation 7cos x + sin x = k has only one solution in the interval 0 x < 360. () [013_R June Q8] 17. (a) Express sin θ 4 cos θ in the form R sin(θ α), where R and α are constants, R Find > 0 and 0 < α <. Give the value of α to 3 decimal places. (b) (i) the maximum value of H(θ), H(θ) = 4 + 5(sin 3θ 4cos3θ) (ii) the smallest value of θ, for 0 θ π, at which this maximum value occurs. Core Maths 3 Trigonometry Page 38

39 Find (c) (i) the minimum value of H(θ), 18. (ii) the largest value of θ, for 0 θ π, at which this minimum value occurs. [014 June Q9] Figure 1 Figure 1 shows the curve C, with equation y = 6 cos x +.5 sin x for 0 x π. (a) Express 6 cos x +.5 sin x in the form R cos(x α), where R and α are constants with R > 0 and 0 < α <. Give your value of α to 3 decimal places. (b) Find the coordinates of the points on the graph where the curve C crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 5 weeks later. She models her results with the continuous function given by H t t 1 6cos.5sin 5 5, 0 t 5 where H is the number of hours of daylight and t is the number of weeks since her first recording. Use this function to find (c) the maximum and minimum values of H predicted by the model, (d) the values for t when H = 16, giving your answers to the nearest whole number. (6) [014_R June Q7] Core Maths 3 Trigonometry Page 39

40 19. g( ) = 4 cos + sin. Given that g( ) = R cos ( ), where R > 0 and 0 < < 90, (a) find the exact value of R and the value of to decimal places. (b) Hence solve, for 90 < < 90, 4 cos + sin = 1, giving your answers to one decimal place. Given that k is a constant and the equation g( ) = k has no solutions, (c) state the range of possible values of k. () [015, June Q3] 0. (a) Express cos θ sin θ in the form R cos (θ + α), where R and α are constants, R > 0 and 0 < α < 90 Give the exact value of R and give the value of α to decimal places. (b) Hence solve, for 0 θ < 360, 15. cos sin 1 Give your answers to one decimal place. (c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of θ for which 15. cos sin 1 Give your answer to one decimal place. () [016, June Q3] Core Maths 3 Trigonometry Page 40

41 1. (a) Write 5 cos θ sin θ in the form R cos (θ + α), where R and α are constants, R > 0 and 0 α < π Give the exact value of R and give the value of α in radians to 3 decimal places. (b) Show that the equation can be rewritten in the form 5 cot x 3 cosec x = 5 cos x sin x = c where c is a positive constant to be determined. (c) Hence or otherwise, solve, for 0 x < π, 5 cot x 3 cosec x = giving your answers to decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.) () [017, June Q4] Core Maths 3 Trigonometry Page 41

42 . (a) Express sin θ + cos θ in the form Rsin (θ + α), where R and α are constants, R > 0 and 0 < α < 90. Give your value of α to decimal places. Figure 4 Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, C, D and E, each of which is in contact with two horizontal parallel lines l1 and l. Rectangle D touches rectangles C and E as shown in Figure 4. Rectangles C, D and E each have length 4 m and width m. The acute angle θ between the line l and the longer edge of each rectangle is shown in Figure 4. Given that l1 and l are 4 m apart, (b) show that Given also that 0 < θ < 45, (c) solve the equation sin θ + cos θ = sin θ + cos θ = () giving the value of θ to 1 decimal place. Rectangles C and D and rectangles D and E touch for a distance h m as shown in Figure 4. Using your answer to part (c), or otherwise, (d) find the value of h, giving your answer to significant figures. [014, June, IAL, Q13] Core Maths 3 Trigonometry Page 4

43 3. (a) Express 10 cos θ 3 sin θ in the form R cos (θ + α), where R > 0 and 0 < α < 90 Give the exact value of R and give the value of α to decimal places. Alana models the height above the ground of a passenger on a Ferris wheel by the equation H = 1 10 cos (30t) + 3 sin (30t) where the height of the passenger above the ground is H metres at time t minutes after the wheel starts turning. (b) Calculate (i) the maximum value of H predicted by this model, (ii) the value of t when this maximum first occurs. Give each answer to decimal places. (c) Calculate the value of t when the passenger is 18m above the ground for the first time. Give your answer to decimal places. (d) Determine the time taken for the Ferris wheel to complete two revolutions. () [015, Jan, IAL, Q13] 4. (a) Express 1.5sin θ 1.cos θ in the form Rsin (θ α), where R > 0 and 0 < α <. Give the value of R and the value of α to 3 decimal places. The height, H metres, of sea water at the entrance to a harbour on a particular day, is modelled by the equation t H = sin 6 t 1.cos, 0 t < 1 6 where t is the number of hours after midday. (b) Using your answer to part (a), calculate the minimum value of H predicted by this Core Maths 3 Trigonometry Page 43

44 model and the value of t, to decimal places, when this minimum occurs. (c) Find, to the nearest minute, the times when the height of sea water at the entrance to the harbour is predicted by this model to be 4 metres. (6) [015, June, IAL, Q11] 5. (a) Express 3sinx + 5cosx in the form R sin (x + α), where R > 0 and 0 < α < Give the exact value of R and give the value of α to 3 significant figures. (b) Solve, for 0 < x < π, 3sinx + 5cosx = 4 g(x) = 4(3sinx + 5cosx) + 3 (c) Using your answer to part (a) and showing your working, (i) find the greatest value of g(x), (ii) find the least value of g(x). [016, Jan, IAL, Q10] 6. (a) Express 3sinx + 5cosx in the form R sin (x + α), where R > 0 and 0 < α < Give the exact value of R and give the value of α to 3 significant figures. (b) Solve, for 0 < x < π, 3sinx + 5cosx = 4 g(x) = 4(3sinx + 5cosx) + 3 (c) Using your answer to part (a) and showing your working, (i) find the greatest value of g(x), (ii) find the least value of g(x). [016, Jan, IAL, Q10] Core Maths 3 Trigonometry Page 44

45 7. (a) Express 3 cos θ + 5 sin θ in the form R cos(θ α), where R and α are constants, R > 0 and 0 < α < 90. Give the exact value of R and give the value of α to decimal places. (b) Hence solve, for 0 θ < 360, the equation 3 cos θ + 5 sin θ = Give your answers to one decimal place. (c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of θ for which 3 cos θ 5 sin θ = () [016, June, IAL, Q1] 8. (a) Express 35 sin x 1 cos x in the form R sin(x α), where R > 0 and 0 < α < p figures. Give the exact value of R, and give the value of α, in radians, to 4 significant (b) Hence solve, for 0 x p, 70 sin x 4 cos x = 37 (Solutions based entirely on graphical or numerical methods are not acceptable.) 7000 y 31 (35sin x1cos x), x > 0 (c) Use your answer to part (a) to calculate (i) the minimum value of y, (ii) the smallest value of x, x > 0, at which this minimum value occurs. [017, Jan, IAL, Q11] Core Maths 3 Trigonometry Page 45

46 9. (a) Write sin θ cos θ in the form R sin (θ α), where R and α are constants, R > 0 and 0 < α 90. Give the exact value of R and give the value of α to one decimal place. Figure 3 shows a sketch of the graph with equation y = sin θ cos θ, 0 θ < 360 (b) Sketch the graph with equation y = sin θ cos θ, 0 θ < 360 stating the coordinates of all points at which the graph meets or cuts the coordinate axes. The temperature of a warehouse is modelled by the equation f(t) = 5 + sin (15t) cos (15t), 0 t < 4 where f(t) is the temperature of the warehouse in degrees Celsius and t is the time measured in hours from midnight. State (c) (i) (ii) Give the maximum value of f(t), the largest value of t, for 0 t < 4, at which this maximum value occurs. your answer to one decimal place. [017, June, IAL, Q10] Core Maths 3 Trigonometry Page 46