Harmonic form R cos(x ± a)

Express 140 cos θ – 480 sin θ in the form K cos(θ + α)

whereK 0 and 0 < α < 90

State the value of K and give the value of α, in degrees, to 2 decimal places.

A scientist studies the number of rabbits and the number of foxes in a wood for one year.

The number of rabbits, R, is modelled by the equation

where t months is the time after the start of the year and A is a constant.

Given that, during the year, the maximum number of rabbits in the wood is 1500

find a complete equation for this model.

Hence write down the minimum number of rabbits in the wood during the year according to the model.

The actual number of rabbits in the wood is at its minimum value in the middle of April.

Use this information to comment on the model for the number of rabbits.

The number of foxes, F, in the wood during the same year is modelled by the equation

The number of foxes is at its minimum value after T months.

Find, according to the models, the number of rabbits in the wood at time T months.

Express 2 cosθ + 8 sinθ in the form Rcos(θ – α), where R and α are constants,

The first three terms of an arithmetic sequence are

Given that S₉ represents the sum of the first 9 terms of this sequence as x varies,

find the exact maximum value of S₉

deduce the smallest positive value of x at which this maximum value

Express 2cosθ – sinθ in the form Rcos (θ + α), where R > 0 and 0 α Give the exact value of R and the value of α in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel.

The wheel is free to rotate about a fixed axis through the point C.

The point P is at the end of one of the paddles of the wheel, as shown in Figure 6.

The water level is assumed to be horizontal and of constant height.

The vertical height, H metres, of P above the water level is modelled by the equation

H = 3 + 4 cos (0.5 t) – 2 sin (0.5 t)

where t is the time in seconds after the wheel starts rotating.

Using the model, find

the maximum height of P above the water level,

the value of t when this maximum height first occurs, giving your answer to one decimal place.

In a single revolution of the wheel, P is below the water level for a total of T seconds.

According to the model,

find the value of T giving your answer to 3 significant figures.

(Solutions based entirely on calculator technology are not acceptable.)

In reality, the water level may not be of constant height.

Explain how the equation of the model should be refined to take this into account.

Express 2cosθ – sinθ in the form Rcos (θ + α), where R > 0 and 0 α Give the exact value of R and the value of α in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel. The wheel is free to rotate about a fixed axis through the point C. The point P is at the end of one of the paddles of the wheel, as shown in Figure 6. The water level is assumed to be horizontal and of constant height. The vertical height, H metres, of P above the water level is modelled by the equation H = 3 + 4 cos (0.5 t) – 2 sin (0.5 t) where t is the time in seconds after the wheel starts rotating. Using the model, find

the maximum height of P above the water level,

the value of t when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, P is below the water level for a total of T seconds. According to the model,

find the value of T giving your answer to 3 significant figures. (Solutions based entirely on calculator technology are not acceptable.)

In reality, the water level may not be of constant height.

Explain how the equation of the model should be refined to take this into account.

Express sinx + 2 cosx in the form Rsin (x + α) where R and α are constants, R 0

Give the exact value of R and give the value of α in radians to 3 decimal places. The temperature, θ°C, inside a room on a given day is modelled by the equation

where t is the number of hours after midnight. Using the equation of the model and your answer to part (a),

deduce the maximum temperature of the room during this day,

find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

Solve, for 0≤ θ <360°, the equation

giving your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

g() = 4cos2 + 2sin2 Given that g() = R cos(2),where > 0 and 0 < < 90°,

find the exact value of and the value of α to 2 decimal places.

Hence solve, for –90° < < 90°,

giving your answers to one decimal place.

Given that k is a constant and the equation g() = k has no solutions,

state the range of possible values of k.

Figure 1 shows the curve C, with equation y = 6 cos x + 2.5 sin x for 0 ≤ x ≤ 2

Express 6 cos x + 2.5 sin x in the form R cos(x – ), where R and α are constants with R 0 and 0 α . Give your value of α to 3 decimal places.

Find the coordinates of the points on the graph where the curve C crosses the coordinate axes.

A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by

where H is the number of hours of daylight and t is the number of weeks since her first recording. Use this function to find

the maximum and minimum values of H predicted by the model,

the values for t when H = 16, giving your answers to the nearest whole number. [ You must show your working. Answers based entirely on graphical or numerical methods are not acceptable. ]

Express 2 sin – 4 cos in the form R sin( ± ), where R and are constants, R 0

Give the value of to 3 decimal places.

Find

the maximum value of H(),

the smallest value of , for 0 ≤ θ , at which this maximum value occurs.

Find

the minimum value of H(),

the largest value of , for 0 ≤ , at which this minimum value occurs.

find the exact value of R and the value of a to one decimal place. (3)

Hence solve the equation

State the values of & for which the equation

Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s^–1. Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in Figure 2. Kate’s speed is V m s^–1 and she moves in a straight line, which makes an angle , 0 < < 150, with the edge of the road, as shown in Figure 2. You may assume that V is given by the formula

Express 24sin + 7cos in the form Rcos( – ), where R and are constants and where R > 0 and 0 < < 90, giving the value of to 2 decimal places.

Given that varies,

find the minimum value of V.

Given that Kate’s speed has the value found in part (b),

find the distance AB.

Given instead that Kate’s speed is 1.68 m s^–1,

find the two possible values of the angle , given that 0 < < 150.

Express 6 cos+8sin in the form Rcos(), where R 0 and 0 α Give the value of to 3 decimal places.

Calculate

the maximum value of p(),

the value of at which the maximum occurs.

f (x) = 7 cos 2x − 24sin 2x Given that f (x) = R cos(2x + α), where R 0 and 0 α 90,

find the value of R and the value of .

Hence solve the equation

for 0 ≤ x180, giving your answers to 1 decimal place.

Express 14cos² x 48sin x cos x in the form a cos 2x + bsin 2x + c, where a, b, and c are constants to be found.

Hence, using your answers to parts (a) and (c), deduce the maximum value of

Express 2cos 3x – 3sin 3x in the form R cos (3x + ), where R and are constants, R 0 and 0 α . Give your answers to 3 significant figures.

Show that f (x) can be written in the form

where R and are the constants found in part (a).

Hence, or otherwise, find the smallest positive value of x for which the curve with equation y = f (x) has a turning point.

Express 2 sin θ – 1.5 cos θ in the form R sin (θ – α), where R 0 and 0 . Give the value of α to 4 decimal places.

Find the maximum value of 2 sin θ – 1.5 cos θ.

Find the value of θ, for 0 θ π, at which this maximum occurs.

Tom models the height of sea water, H metres, on a particular day by the equation

where t hours is the number of hours after midday.

Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs.

Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

Express 5 cos x – 3 sin x in the form R cos(x + α), where R > 0 and 0 < α < .

Hence, or otherwise, solve the equation 5 cos x – 3 sin x = 4 for 0 x 2›, giving your answers to 2 decimal places.

Use the identity cos(A + B) = cos Acos B sin Asin B, to show that

The curves C₁ and C₂ have equations

Show that the x-coordinates of the points where C₁ and C₂ intersect satisfy the equation

Express 4cos2x + 3sin 2x in the form R cos (2x – α), where R > 0 and 0 < α < 90°, giving the value of α to 2 decimal places.

Hence find, for 0 x 180°, all the solutions of

giving your answers to 1 decimal place.

Express 3 cos θ + 4 sin θ in the form R cos(θ – α), where R and α are constants, R > 0 and 0 < α < 90°.

Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.

The temperature, f (t), of a warehouse is modelled using the equation f (t) = 10 + 3 cos(15 t)° + 4 sin(15 t)°, where t is the time in hours from midday and 0 t < 24.

Calculate the minimum temperature of the warehouse as given by this model.

Find the value of t when this minimum temperature occurs.