Harmonic form R cos(x ± a)

Express 2 sin x + 5 cos x in the form R sin (x + α)

where R > 0 and 0° ≤ α ≤ 90°

In 2010, the water temperature at a beach could be modelled by the formula

T = 14.2 – (2 sin d ° + 5 cos d °)

where: T is the temperature of the water in °C d is the number of days after 1 January 2010

Find the value of d when the temperature of the water reached its lowest value in 2010

Find the maximum temperature of the water predicted by the model in 2010

Give your answer to one decimal place.

Find the number of weeks in 2010 for which the temperature of the water was higher than 15 °C

Give your answer to the nearest whole number.

One of the rides at a theme park is a room where the floor and ceiling both move up and down for 10 seconds.

At time t seconds after the ride begins, the distance f metres of the floor above the ground is

At time t seconds after the ride begins, the distance c metres of the ceiling above the ground is

The ride is shown in the diagram below.

Show that the initial distance between the floor and ceiling is 8 metres.

Show that the distance d metres between the floor and ceiling at time t is given by

where R and are positive constants to be found.

Hence, find the minimum distance between the ceiling and the floor.

Give your answer to the nearest centimetre.

Given that

find the value of R.

Circle your answer.

6 8 10 14

A curve has equation

where a and b are constants. The maximum value of y is 4 and the curve passes through the point 23 as shown in the diagram.

Find the exact values of a and b.

Determine a sequence of transformations which maps the graph of y sin x onto

Fully justify your answer.