Trigonometry

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.

Figure 4 Figure 5 Figure 4 shows a sketch of a Ferris wheel. The height above the ground, Hm, of a passenger on the Ferris wheel, t seconds after the wheel starts turning, is modelled by the equation

where A, b and α are constants. Figure 5 shows a sketch of the graph of H against t, for one revolution of the wheel. Given that

• the maximum height of the passenger above the ground is 50 m

• the passenger is 1 m above the ground when the wheel starts turning

• the wheel takes 720 seconds to complete one revolution

find a complete equation for the model, giving the exact value of A, the exact value of b and the value of α to 3 significant figures.

Explain why an equation of the form

where d is a positive constant, would be a more appropriate model.

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.

The depth of water, D metres, in a harbour on a particular day is modelled by the formula

where t is the number of hours after midnight.

A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.

Find the depth of the water in the harbour when the boat enters the harbour.

Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.)