Trig Fns - Different

Theoretical	1. (i) Differentiate y = sin x + cos x. (ii) Find the minimum point and the point(s) of inflection in the domain 0 ≤ x ≤ 2π.
	2. The diagram shows a point F on the unit circle x² + y² = 1 at an angle of θ to the positive direction of the x-axis (0 ≤ θ ≤ π). The tangent at F is perpendicular to OF. It intersects the x-axis at D and the line y = 1 at G. The line y = 1 intersects the y axis at E. (i) Show that the equation of the line DF is xcos θ+ ysin θ = 1. (ii) Find the length of EG in terms of θ. (iii) Show that the area A of the trapezium EOGD is given by (iv) Find the angle θ which leads to the minimum area of the trapezium. Answer:Area is min at θ = π/6.
	3. Consider the curve y = sin x (1 + cos x) for 0 ≤ x ≤ 2 π. (i) show that (ii) Show that . (iii) Find the stationary points on the curve. (iv) Determine the nature of these stationary points. (v) Sketch the curve in the given domain.
Practical situations	4. After several days of measurement and testing the waves at Curl Curl Beach before the Trial, Charlie determined that wave height could be modelled by the equation where h is the height of the tide at t hours. (i) What was the height of the tide after 8 hours? (ii) Find the rate at which the height of the time was changing at 8 hours. (iii) Was the tide rising or falling at the 8 hour mark. Justify your answer. Answer:(i) 3 m. (ii) rate was 0.9 m/hour. (iii) at the rate in (ii) is positive, the height is increasing.
	5. The diagram above shows a rhombus ABCD with <BAD = α (0 < α < π/2). The sides are each 5 cm long. BD is an arc of a circle centered at A. (i) Show that the area of the coloured region can be expressed as (ii) Find the maximum area of the coloured region.
	6. Two sides of a triangle are of lengths 8 cm and 12 cm. The angle between the two sides is θ. (i) Write down an expression for the area (A) of this triangle in terms of θ. (ii) Find the greatest area of the triangle. Answer:Max area is 48 cm².
Type 2: 2D shapes - sectors.	7. AOB is the sector of a circle with centre at O and a radius r. The angle subtended at O by the arc AB is . CDEF is a rectangle drawn in the sector and ∠EOF = α. (i) Show that . (ii) Given that , show that the area of rectangle CDEF can be expressed as . (iii) Find the value of α which will produce the rectangle of maximum area. Answer.(iii) α = π/6.
	8. In the diagram below, a truck's windscreen wiper blade sweeps across the region ABCD where BC and AD are arcs of circles with centre O. The intervals OA and AB are x cm and 3x cm respectively with < BOC = θ. The perimeter of the shaded region ABCD is 240 cm. (i) Determine an equation for θ in terms of x. (ii) Show that the area ABCD is A = 9x (40 - x). (iii) Find the maximum area of the shaded region ABCD. Answer.Max area = 3,600 cm².
	9. A sector with an area of 100 cm² has a radius r and an angle of θ radians. (i) Show that the perimeter P cm of the sector can be expressed as (ii) If r and θ vary in such a way that the area of the sector remains constant at 100 cm², find the radius and the value for θ which give the smallest value for the perimeter. Answer.r = 10 cm and θ = 2 rads.
Type 3: 3D shape.	10. A gutter is made out of sheet metal 3a units wide by bending it as shown in the diagram. The length of the gutter is b units. (i) Show that the volume of the gutter can be expressed as (ii) Show that . (iii) Determine the value of θ so that the gutter has a maximum volume. Answer.Max volume at θ = π/6 rads.
Type 4: Rates	11. Evie is at E, shown in the diagram below. At that point, Evie is 100 metres from Point B on the beach. She decides she wants to use her surfboard which she left at S which is further down the beach and 200 metres from where Evie is now treading water. Evie is a reasonably good athlete. She swims at 2 m/sec and can run at 6 m/sec. (i) Show that where θ =∠BEQ. (ii) Show that the total time of travel T (in seconds) for Evie to swim and run from E to S via Q can be expressed as (iii) In which direction θ should Evie start swimming to reach her surfboard in the shortest time? (iv) What is the shortest time for Evie to reach her surfboard? Answer.(iii) Direction is 19^o 28'. (iv)Time is 1 min 36 secs.