THE INVERSE TRIGONOMETRIC FUNCTIONS

Transcription

1 THE INVERSE TRIGONOMETRIC FUNCTIONS

2 Question 1 (**+) Solve the following trigonometric equation ( x ) π + 3arccos + 1 = 0. 1 x = Question (***) It is given that arcsin x = arccos y. Show, by a clear method, that x + y = 1. C3U, proof Question 3 (***) Solve the following trigonometric equation ( x ) 3arccot 3 π = 0. x = 4 3 3

3 Question 4 (***+) A curve C is defined by the equation ( x ) y = arcsin 1, 0 x. a) Describe the geometric transformations that map the graph of arcsin x onto the graph of C. b) Sketch the graph of C. The sketch must include the coordinates of any points where the graph of C meets the coordinate axes and the coordinates of the endpoints of C. C3P, translation by 1 unit to the right, followed by reflection in the x axis Question 5 (***+) Simplify, showing all steps in the calculation, the following expression tan ( arctan 3 arctan ), giving the final answer as an exact fraction. 1 7

4 Question 6 (***+) Show clearly that if x > 0 1 π arctan x + arctan =. x proof Question 7 (***+) Solve the equation showing clearly all the workings. arctan 1 ( ) = arccos x, x = 3 5

5 Question 8 (***+) Simplify, showing all steps in the calculation, the expression tan arctan 1 + arctan 1 3 4, giving the final answer as an exact fraction Question 9 (***+) Show clearly that arccos 4 ( ) arccos 7 ( ) =. 5 5 proof

6 Question 10 (***+) Show clearly that arctan + arctan 5 = arctan proof Question 11 (***+) Show clearly that ( x) = sin arctan x x + 1. proof

7 Question 1 (***+) Prove the trigonometric identity π arcsin x + arccos x =. proof Question 13 (***+) Show clearly that arctan 1 + arctan 4 = arctan proof

8 Question 14 (***+) Solve the trigonometric equation arcsin x = arccos x. x = 1 5 Question 15 (***+) Using a detailed method, show that arctan 1 arctan = 4 π. proof

9 Question 16 (***+) Show, by detailed workings, that 3π arctan + arctan 3 =. 4 proof

10 Question 17 (***+) Use a detailed method to show that arccos 5 + arccos 10 = π. 4 SYN-B, proof Question 18 (***+) Find the general solution of the following trigonometric equation arctan sin ( x) arctan ( sec x) =. π SYNF-B, x = + kπ, k Z 4

11 Question 19 (****) y1 = arcsin x y O y B y = arccos x C D x A O E x The diagrams above shows the graphs of y1 = arcsin x and y = arccos x. The graph of y 1 has endpoints at A and B. The graph of y has endpoints at C and E, and D is the point where the graph of y crosses the y axis. a) State the coordinates of A, B, C, D and E. The graph of y can be obtained from the graph of y 1 by a series of two geometric transformations which can be carried out in a specific order. b) Describe the two geometric transformations. c) Deduce using valid arguments that arcsin x + arccos x = constant, stating the exact value of this constant. π A 1,, B 1, π, C ( 1, π ), D 0, π, ( 1,0 ) E, constant π =

12 Question 0 (****) y = arcsin x, 1 y 1. a) Show that dy dx = 1 1 x. The point P 1 (, ) 6 k, where k is a constant lies on the curve with equation π arcsin 3x + arcsin y =, x 1, y 1. 3 b) Find the value of the gradient at P. SYNF-A,

13 Question 1 (****) f ( x) sin x 1 a) Find an expression for f ( x) 1 b) Sketch the graph of f ( x) = +, π x π... The sketch must include the coordinates of any points where the graph of f 1 ( x ) meet the coordinate axes as well as the coordinates of its endpoints. ( ) = arcsin ( 1) 1 1 f x x Question (****) Solve the following trigonometric equation 3 tan ( arctan 3x arctan ) + tan ( arctan 3 arctan x) =. 8 x = 1

14 Question 3 (****) A curve has equation ( x ) y = π arccos + 1, x 0. a) Describe geometrically the 3 transformations that map the graph of y = arccos x, 1 x 1, onto the graph of y = π arccos x + 1, x 0. ( ) b) Sketch the graph of ( x ) y = π arccos + 1, x 0. The sketch must include the coordinates of any points where the graph meets the coordinate axes. c) Use symmetry arguments to find the area of the finite region bounded by ( x ) y = π arccos + 1, x 0, and the coordinate axes. C3L, translation by 1 unit to the right, followed by reflection in the x axis, area = π

15 Question 4 (****) Solve the following trigonometric equation 1 1 π arctan + arctan =. x x x = 1,

16 Question 5 (****) ( ) 1 ( ) 1 a) Find an expression for f ( x) 1 b) Sketch the graph of f ( x) The sketch must include f x = + tan x, π x π. 1 the equations of the asymptotes of f ( x).. 1 the coordinates of any points where the graph of f ( x) coordinate axes. meets the ( ) = arctan ( + 1) 1 1 f x x

17 Question 6 (****) Solve the following trigonometric equation 3 6x arctan = arcsin x 5. x = ± 4

18 Question 7 (****) The curves C 1 and C have respective equations = ( x ) and y ( x ) y1 3arcsin 1 1 = arccos 1. a) Sketch in the same diagram the graphs of C 1 and C. The sketch must include the coordinates of any points where the graphs of C 1 and C meet the coordinate axes as well as the coordinates of the endpoints of the curves. b) Use a suitable iteration formula of the form x f ( x ) x = to find n+ 1 = n with the x coordinate of the point of intersection between the graphs of C 1 and C. x 1.59

19 Question 8 (****) Make x the subject of the equation ( ) ( ) arctan 1+ x + arctan 1 x = y. x = ± tan y

20 Question 9 (****) It is given that d du ( arcsin u) = 1 1 u, u 1. Hence show that if 1 y = sin arcsin x, then a) ( x ) dy 1 4 = 1 y dx b) ( ) 1 4 d y dy x 4x y 0 dx dx + =.. FP-G, proof

21 Question 30 (****) y = arcsin x, 1 x 1. a) By expressing arccos x in terms of y, show that π arcsin x + arccos x =. b) Hence, or otherwise, solve the equation ( x ) ( x ) 3arcsin 1 = arccos 1. π ( ) x = 1+ sin

22 Question 31 (****) A curve has equation y = arcsin x, 1 1 x, π π y. a) By finding dx dy and using an appropriate trigonometric identity show that dy dx = 1 4x. b) Show further that i. d y dx = Ax ( ) 3 1 4x, 3 d y Bx + C ii. = 3 dx 1 4x ( ) 5, where A, B and C are constants to be found. SYN-B, proof

23 Question 3 (****) y = arcsin x, 1 1 x, π π y. a) By finding dx dy and using an appropriate trigonometric identity show that dy dx = 1 1 x. A curve C has equation y = x arcsin x, 1 1 x, π π y. b) Find the exact value of dy dx at the point on C where x = 1. 4 FP-A, ( ) π +

24 Question 33 (****) = arcsin + 1 4, 1 x 1 y x x x. Show clearly that 3 d y dy d y y x x 3 = dx dx dx. FP-K, proof

25 Question 34 (****) Use trigonometric algebra to solve the equation sin arcsin 1 + arccos x = 1. 4 x = 1 4 Question 35 (****) The curve C has equation ( x ) y = arcsin 1, 0 x 1. Find the coordinates of the point on C, whose gradient is. ( ) 1,0

26 Question 36 (****) Find a simplified expression for d arctan dx x 4 x FP-Q, d x 1 arctan = dx 4 x 4 x Question 37 (****) Solve the following trigonometric equation. arctan x + arctan x = arctan 3, x R. SP-F, x = 1

27 Question 38 (****+) sin 3θ 3sinθ 4sin θ. 3 a) Prove the validity of the above trigonometric identity by considering the sin θ + θ. expansion of ( ) b) Hence or otherwise solve the equation 1 ( ) arcsin x = 3arcsin. 3 x = 3 7 Question 39 (****+) Solve the following simultaneous equations arctan x + arctan y = arctan 8 x + y =. x = 1, y = 3, in either order

28 Question 40 (****+) π π = 0 x < < x π. f ( x) sec x 1 a) Sketch in the same diagram the graphs of f ( x ) and ( ) 1 b) State the domain and range of ( ) c) Show clearly that arcsec x arccos 1 ( ) f x = arcsec x. = x. f x = arcsec x. d 1 dx = x x d) Show further that ( arcsec x) 4., 1 range: 0 f ( x) π, f 1 ( x) domain: x 1 x 1 π

29 Question 41 (****+) Show clearly that 3 1 arctan + arctan = π. 5 proof Question 4 (****+) Show clearly that 1 x π arctan x + arctan =. 1 + x 4 proof

30 Question 43 (****+) Solve the following trigonometric equation. x 5 x 4 π arctan + arctan =, x R. x 1 x 3 4 x = 3, x = 6

31 Question 44 (****+) y = arccos x, 1 x 1, 0 y π. a) By writing y = arccos x as x = cos y, show that dy dx = 1 1 x. The curve C has equation ( ) y = arccos x 1ln 1 x, x > 0. b) Show that the y coordinate of the stationary point of C is 1 ( ln 4) 4 π +. FP-O, proof

32 Question 45 (****+) Solve the following trigonometric equation arcsin x + arccos 3 = arctan x = Question 46 (****+) Find the solution of the equation 1 x 1 arctan = arctan x. 1+ x SP-C, x = 3 3

33 Question 47 (****+) The functions f and g are defined by f ( x) 3sin x, x R, ( ) g x 1 a) Find an expression for f g ( x) 1 b) Determine the domain of f g ( x) 6 3x, x R. SP-C, f 1 g ( x) arcsin ( x ).. π π x =, 3 x 1 or 1 x 3

34 Question 48 (****+) y = arctan x, x R. a) By writing the above equation in the from x g ( y) The function f is defined as d dx 1 =. 1 + x ( arctan x) =, show that f ( x) = arctan x, x R, x 0. b) Show further that 1 3 ( ) f x = x ( 3x + 1)( x + 1). 4 FP-M, proof

35 Question 49 (****+) arctan + arctan = arctan x 3 x Show that x = 5 is one of the solutions of the above trigonometric equation, and find in exact surd form the other two solution. 10 ± x = 31

36 Question 50 (****+) y = arccos x, x R, 1 x 1. a) By writing the above equation in the from x = f ( y), show that d dx ( arccos x) = 1 1 x. A curve has equation ( x ) y = arccos 1, x R, 0 < x. b) Show further that d y dx = x ( ) 3 x. c) Show clearly that 3 5 ( x ) 16 d y 4 d y dy x dy = dx dx dx dx. FP-U, proof

37 Question 51 (*****) Given the simultaneous equations 3tanθ + 4 tanϕ = 8 π θ + ϕ =, find the possible value of tanθ and the possible value of tanϕ. [ tan,tan ] =, 1 =, 3 θ ϕ 3

38 Question 5 (*****) Simplify, showing all steps in the calculation, the expression giving the answer in terms of π. 4 arctan arctan arctan 3 3 +, π 4

39 Question 53 (*****) 1 x y = arctan x + arctan 1 + x, x R. Without simplifying the above expression, use differentiation to show that for all values of x dy 0 dx =. proof

40 Question 54 (*****) A curve C has equation arctan x y = e, x R. a) Show, with detailed workings, that 3 d y = dx ( x x ) ( 1+ x ) arctan x e 3 3. b) Deduce that C has a point of inflection, stating its coordinates. SPX-J, 1,e arctan 1

41 Question 55 (*****) Solve the following trigonometric equation 1 ( ) ( x) 1 ( x) cos arcsin sin arccos = 4, x R. 4 4 SPX-L, x = 1 4

42 Question 56 (*****) Simplify, showing all steps in the calculation, the expression arctan8 + arctan + arctan, 3 giving the answer in terms of π. SP-A, π

43 Question 57 (*****) It is given that α + cos x y = arcsin 1 + α cos x, where α is a constant. Show that dy 1 α =. dx 1 + α cos x SPX-H, proof

44 Question 58 (*****) The functions f and g are defined by f ( x) cos x, x R, 0 x π ( ) 1 x g x a) Solve the equation fg ( x ) = 1., x R. 1 b) Determine the values of x for which f g ( x) is not defined. π x = ± 1, x < or x > 6

45 Question 59 (*****) The acute angles θ and ϕ satisfy the following equations cosθ = cosϕ sinθ = 3sinϕ. Show clearly that θ + ϕ = π arctan 15 C3O, proof

46 Question 60 (*****) Show clearly that 4 4arccot + arctan = π. 7 C3T, proof

47 Question 61 (*****) Solve the following trigonometric equation 5π arcsin x + arccos x =. 6 SP-B, x = 1

48 Question 6 (*****) Find the only finite solution of the equation x 1 π arcsin + arctan =. x 1 x + 1 x = 0

49 Question 63 (*****) Solve the trigonometric equation 1 x π arctan ( x ) + arcsin =, x R. 1 + x SP-S, x = 4

50 Question 64 (*****) Use trigonometric algebra to fully simplify arctan 1+ sin x 1 sin x π, 0 < x <, 1+ sin x + 1 sin x 4 giving the final answer in terms of x. SP-R, 1 x

51 Question 65 (*****) Use trigonometric algebra to solve the equation π arctan x + arccot x =. 3 You may assume that arccot x is the inverse function for the part of cot x for which 0 x π. SP-U, x = 3

52 Question 66 (*****) Use trigonometric algebra to fully simplify arctan + arccos arctan 5 +, 5 8 giving the final answer in terms of π. SP-Q, π 4

53 Question 67 (*****) 1 f ( x) = arctan ( 3x) + arcsin, x R. 9x + 1 Show, by a detailed method, that =. a) f ( x) 0 b) arctan ( 3x) 1 + arcsin kπ, stating the value of the constant k. 9x + 1 FP-S, k = 1

54 Question 68 (*****) It given that π arctan x + arctan y + arctan z =. Show that x, y and z satisfy the relationship xy + yz + zx = 1. SP-Y, proof

55 Question 69 (*****) Use a trigonometric algebra to solve the following equation π + =. 8 ( arctan x) ( arccot x) 5 You may assume that y = arccot x is the inverse function of y = cot x, 0 x π SP-Z, x = 1

56 Question 70 (*****) Solve the following trigonometric equation ( ) arctan x cos arcsin 1 x = 1 π. 4 SP-V, x = 1, x =

57 Question 71 (*****) On a clearly labelled set of axes, draw a detailed sketch of the graph of ( arcsin ) y = x arccos x, 1 x 1. SP-N, graph

58 Question 7 (*****) Solve the following trigonometric equation ( x + ) = ( x) sin arccot 1 cos arctan. You may assume that y = arccot x is the inverse function for y = cot x, 0 x π. SP-W, x = 1

59 Question 73 (*****) It is given that ( arcsin ) ( arccos ) x + x = kπ, x 1, for some constant k. a) Show that a necessary but not sufficient condition for the above equation to have solutions is that k 1. 3 b) Solve the equation given that it only has one solution. c) Given instead that that k = 7, find the two solutions of the equation, giving 96 x = sin aπ, where a Q. the answers in the form ( ) SP-X, x =, π 5π x = sin, x = sin 1 1

60 Question 74 (*****) Sketch the graph of ( ) arcsin ( cos ) f x = x, in the largest domain that the function is defined. Indicate the coordinates of any intersections with the axes, and the coordinates of the cusps of the curve. SP-V, graph

61 Question 75 (*****) x y = arctan 1 x, x R. Differentiate y with respect to x arcsin 1 + x, fully simplifying the answer. FP-T, 1

62 Question 76 (*****) 1 x Differentiate arctan with respect to x the graph of the resulting gradient function. arccos x 1 x, and hence sketch 1 sign SPX-V, ( x)

63 Question 77 (*****) It is given that arctan + arctan A + arctan B = π. It is further given that A and B are distinct positive real numbers other than unity. Determine a pair of possible values for A and B. SP-F, 7 5 & 9

64 Question 78 (*****) Prove that for all x such that 1 x 1 ( ) arccos arccos 1 π x + x + 3 3x =. 3 SP-T, proof

65 Question 79 (*****) Find, in exact surd form, the only real solution of the following trigonometric equation π arcsin ( x 1) arccos x =. 6 The rejection of any additional solutions must be fully justified. SP-P, x =

66 Question 80 (*****) By considering the trigonometric identity for tan ( A B), with A arctan ( n 1) B = arctan( n), sum the following series = + and 1 arctan. n + n + 1 n= 1 You may assume the series converges. SPX-L, π 4