(e) (i) Prove that C(x) = C( x) for all x. (2)

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1 Revision - chapters and 3 part two. (a) Sketch the graph of f (x) = sin 3x + sin 6x, 0 x. Write down the exact period of the function f. (Total 3 marks). (a) Sketch the graph of the function C ( x) cos x cosx (c) (d) for x Prove that the function C(x) is periodic and state its period. For what values of x, x, is C(x) a maximum? Let x = x 0 be the smallest positive value of x for which C(x) = 0. Find an approximate value of x 0 which is correct to two significant figures. (e) (i) Prove that C(x) = C( x) for all x. Let x = x be that value of x, < x <, for which C(x) = 0. Find the value of x in terms of x 0. (Total 6 marks) 3. Solve sin x = tan x, where x. (Total 3 marks) 4. A system of equations is given by cos x + cos y =. sin x + sin y =.4. (a) For each equation express y in terms of x. Hence solve the system for 0 x <, 0 < y <.

2 Revision - chapters and 3 part two 5. The angle satisfies the equation tan + cot = 3, where is in degrees. Find all the possible values of lying in the interval [0, 90 ]. 6. Find all the values of θ in the interval [0, ] which satisfy the equation cos = sin. 7. The depth, h (t) metres, of water at the entrance to a harbour at t hours after midnight on a particular day is given by t h (t) = sin,0t 4. 6 (a) Find the maximum depth and the minimum depth of the water. Find the values of t for which h (t) 8. No calculator! 8. The graph below represents y = a sin (x + b) + c, where a, b, and c are constants. y 4 3 3, 4 0 x 3 4 Find values for a, b and c. 9. The angle θ satisfies the equation tan θ 5 sec θ 0 = 0, where θ is in the second quadrant. Find the exact value of sec θ. 0. Solve tan π π =, in the interval θ.

3 Revision - chapters and 3 part two. The diagram below shows the boundary of the cross-section of a water channel. x The equation that represents this boundary is y = 6 sec 36 measured in cm. 3 where x and y are both The top of the channel is level with the ground and has a width of 4 cm. The maximum depth of the channel is 6 cm. Find the width of the water surface in the channel when the water depth is 0 cm. Give your answer in the form a arccos b where a, b. ] π. The function f with domain 0, is defined by f (x) = cos x + 3 sin x. This function may also be expressed in the form R cos (x ) where R > 0 and 0 < α < (a) Find the exact value of R and of α. (i) Find the range of the function f. State, giving a reason, whether or not the inverse function of f exists. (c) Find the exact value of x satisfying the equation f (x) =. (d) Using the result sec xdx = lnsec x + tan x+ C, where C is a constant, show that π 0 dx f ( x) ln(3+ 3). 3

4 Revision - chapters and 3 part two (Total 6 marks) 3. Let θ be the angle between the unit vectors a and b, where 0 < θ < π. Express a b in terms of sin. 4. The function f is defined on the domain [0, ] by f ( ) = 4 cos + 3 sin. (Total 3 marks) (a) Express f ( ) in the form R cos ( ) where 0 < < Hence, or otherwise, write down the value of for which f ( ) takes its maximum value. 5. (a) Express 3 cos sin in the form r cos ( + ), where r > 0 and 0 < α < π, giving r and α as exact values. Hence, or otherwise, for 0, find the range of values of 3 cos sin. (c) Solve 3 cos sin = l, for 0, giving your answers as exact values. (Total 0 marks) 6. Prove that sin 4 cos cos cos4 = tan, for 0 < < π, and 4 (Total 5 marks) 7. (a) Show that cos (A + B) + cos(a B) = cos A cos B Let T n (x) = cos (n arccos x) where x is a real number, x [, ] and n is a positive integer. (i) Find T (x). Show that T (x) = x. (c) (i) Use the result in part (a) to show that T n+ (x) + T n (x) = xt n (x). Hence or otherwise, prove by induction that T n (x) is a polynomial of degree n. () (Total 9 marks) 8. Given that a sin 4x + b sin x = 0, for 0 < x <, find an expression for cos x in terms of a and b. 4

5 Revision - chapters and 3 part two 9. Let z = cos + i sin, for 4 π 4 (a) (i) Find z 3 using the binomial theorem. Use de Moivre s theorem to show that cos 3 = 4 cos 3 3 cos and sin 3 = 3 sin 4 sin 3. (0) Hence prove that sin 3θ sin θ cos3θ cosθ = tan. (6) (c) Given that sin = 3, find the exact value of tan 3. (Total marks) 0. Let Â, Bˆ, Ĉ be the angles of a triangle. Show that tan  + tan Bˆ + tan Ĉ =  tan Bˆ tan Ĉ.. The function f is defined by f (x) = cosec x + tan x. π π (a) Sketch the graph of f for x. Hence state (i) (iii) the x-intercepts; the equations of the asymptotes; the coordinates of the maximum and minimum points. (8) (c) Show that the roots of f (x) = 0 satisfy the equation cos 3 x cos x cos x + = 0. Show that the x-coordinates of the maximum and minimum points on the curve satisfy the equation 4 cos 5 x 4 cos 3 x + cos x + cos x = 0. (8) (d) Show that f ( x) + f ( + x) = 0. (4) (Total 5 marks). In the diagram below, AD is perpendicular to BC. CD = 4, BD = and AD = 3. Cˆ AD = and Bˆ AD =. 5

6 Revision - chapters and 3 part two Find the exact value of cos ( ). 3. Find, in its simplest form, the argument of (sin + i ( cos )) where is an acute angle. (Total 7 marks) 6