Geometry and Sin
1 (a) Show that = tan θ. (b) Hence find the value of cot in the form a + , where a, b . 2 If x satisfies the equation , show that 11 tan x = a + b, where a, b +. 3 The graph below shows y = a cos (bx) + c. Find the value of a, the value of b and the value of c. 4 The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.
The points P and Q lie on the larger circle and angle POQ = x, where 0 < x < π2. Show that the area of the shaded region is 8 sin x – 2x. 6 In the diagram below, AD is perpendicular to BC. CD = 4, BD = 2 and AD = 3. = and = . Find the exact value of cos ( − ). 8 Let sin x = s. (a) Show that the equation 4 cos 2x + 3 sin x cosec3 x + 6 = 0 can be expressed as
8s4 – 10s2 + 3 = 0. (b) Hence solve the equation for x, in the interval [0, π]. 9 A particle P moves in a straight line with displacement relative to origin given by s = 2 sin (πt) + sin(2πt), t ≥ 0, where t is the time in seconds and the displacement is measured in centimetres. (i) Write down the period of the function s. 10 The diagram below shows a curve with equation y = 1 + k sin x, defined for 0 ≤ x ≤ 3π. The point A lies on the curve and B(a, b) is the maximum point. (a)Show that k = –6. (b)Hence, find the values of a and b.
11 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 h (t) = 8 + 4 sin
The points P and Q lie on the larger circle and angle POQ = x, where 0 < x < π2. Show that the area of the shaded region is 8 sin x – 2x. 6 In the diagram below, AD is perpendicular to BC. CD = 4, BD = 2 and AD = 3. = and = . Find the exact value of cos ( − ). 8 Let sin x = s. (a) Show that the equation 4 cos 2x + 3 sin x cosec3 x + 6 = 0 can be expressed as
8s4 – 10s2 + 3 = 0. (b) Hence solve the equation for x, in the interval [0, π]. 9 A particle P moves in a straight line with displacement relative to origin given by s = 2 sin (πt) + sin(2πt), t ≥ 0, where t is the time in seconds and the displacement is measured in centimetres. (i) Write down the period of the function s. 10 The diagram below shows a curve with equation y = 1 + k sin x, defined for 0 ≤ x ≤ 3π. The point A lies on the curve and B(a, b) is the maximum point. (a)Show that k = –6. (b)Hence, find the values of a and b.
11 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 h (t) = 8 + 4 sin