Trigonometric Functions & Graphs (DP IB Maths: AA HL)

A graph has the equation  for the interval  

Sketch the graph on the axes below.

A straight line with equation  intersects the graph of . 

Sketch the line on to the same set of axes.

Find the coordinates of the points of intersection between and  .

An average heart contains a volume of approximately 140 millilitres and pumps out one half of its volume with each beat. A healthy adult has a heart rate of about 70 beats per minute.

Assuming that the heart starts at full capacity, the volume of blood, , in the heart can be modelled as a function of the time, , in seconds.

Write down a model for the volume of blood, , giving your answer in the form , where A, B and C are constants to be found.

Show that the heart is at its minimum capacity at seconds.

A snake moves along a horizontal surface following a line that is 3 m from, and parallel to, the edge of a straight section of river, as shown in the diagram below.

The perpendicular distance of the tip of the snake’s tail from the edge of the river,  cm, can be modelled by the function

for the interval  

where is the horizontal distance, in cm, moved by the tip of the tail from the start of the movement.

A stone is located at a perpendicular distance of 294 cm from the river when the snake has travelled  of the total horizontal distance.

Show that the tip of the tail will collide with the stone.

A hamster runs in its exercise wheel, rotating the wheel at a constant speed. The wheel has a diameter of 14 centimetres and the top of the wheel is positioned at a height of  centimetres above the floor of the cage.

A point at the top of the wheel is marked before the hamster starts to run, turning the wheel clockwise. The hamster takes 4 seconds to turn the wheel one complete revolution.

After seconds, the height of the mark on the wheel above the floor of the cage is given by

  for  

After 26 seconds, the mark is 3 cm above the cage floor. Find .

Find the value of 

Find the value of when the mark is 8 cm above the floor of the cage for the 5^th time in the given time period and state whether the mark is ascending or descending.

A student sets up an experiment with a model car moving along a horizontal surface in a straight line. The car has a trailer attached by a pin joint. A momentary force is applied to the end of the trailer, perpendicular to the direction of travel, which causes it to move sideways back and forth as the car continues to move forwards. A diagram can be seen below.

Point P is situated at the midpoint of the end of the trailer. The displacement, in cm, of point P relative to the centre line of the car in the direction of motion can be modelled by the function

where  is the time in seconds since the application of the force.

Find the total distance that point P has moved perpendicular to the line of motion, in the first 3 seconds.

When the maximum displacement from the centre line does not exceed 2.5 mm, the trailer is considered to be stable.

State the time after which the trailer can be considered stable.

Let   for .

Let .

The function  can be written in the form .

The range of  is . Find  and .

Find the range of g.

The equation  has two solutions where . Find both solutions.

Let 

The diagram below shows the graph of , for 

The first local minimum is at point A  and the next local maximum is at point B .

Find the value of

.

Let 

Find the two solutions for , for .

Let 

Sketch the graph of  for the interval  . Label the coordinates of the local maxima and minima. The coordinates should be given as exact values.

A particle is travelling horizontally whilst moving in and out of a body of water at a constant speed. The particle reaches a maximum height of 1.3 m above the water level and and a depth of 2.6 m.

The particle starts at a depth of  m and takes  seconds to move up through the water, reach the maximum height, dive to the minimum depth and return to its starting depth.

Write down a model for the height,  m, of the particle, relative to water level, at a time  seconds, in the form , where  and  are constants to be found.

A section of a model railway track has a series of rises and falls. The vertical displacement of a train carriage moving along the track,  in m, relative to the horizontal floor, can be modelled by

, for 

where  is the time in minutes.

Find the average vertical speed of the train carriage when it experiences the maximum change in height within the given section of track.

Vertical metal supports are required to ensure that the track is stable. A support is required at either end of the track, as well as at each local maximum and minimum.

Given that there is 7.9 m of metal available to create the supports, show that this is not sufficient to place the supports in the locations required.

A support attached to the side of the track is also required. The support must attach to the model at four separate points, including the end point of the section of the track and can be modelled by the function , where  is a constant.

Find the length of the side support from the first point of connection with the track to the fourth.

The diagram below shows a circle with centre O and radius 2 cm. Points A and B lie on the circumference of the circle and angle , where 

The tangents to the circle at points A and B intersect at point C.

Find the value of  when the shaded area is equal to the area of sector OAB.

A particle, A, starts at a fixed point, O, before being set into motion. The vertical displacement of the particle,   cm, from point O can be modelled by the equation

,   for  

where  is the horizontal displacement, in cm, of the particle from O.

Find the straight line distance of the particle from O at end of the motion.

A second particle, B, starts moving at the same time as particle A. The motion of particle B can be described by the function

,   for   ,   

Given that the particles stop moving at the point of collision, find the value of .

A Ferris wheel with centre O and diameter 100 metres, comprises 32 individual compartments and rotates clockwise at a constant speed of 0.96 kilometres per hour.

Passengers board the Ferris wheel at point A. The height,  metres, of a compartment above the ground after it passes through point A is modelled by the function

, for 

where  is the time elapsed in minutes.

Find the height of point A above the ground.

Points B and C indicate the edges of the region in which a person gains the best view of the city during the rotation of the Ferris wheel. B is reached 10 mins after boarding the wheel. Point C is located at a vertical distance of 26.2 m below point B.

Find

Find the angle .

A large clock face is mounted on a tower, with the centre of the clock face at a height of 19 m above ground level and the tip of the minute hand reaching the circumference of the clock face.

The clock is started at 12 pm and the tip of the minute hand travels a total distance of 5.8 m from its initial position in 35 minutes.

Find a model for the height of the tip of the minute hand,  m, above ground level, in  the form , where is the angle measured clockwise between the number 12 and the minute hand and and  are constants to be found.

Given that the minute hand has travelled a total distance of 31.8 m before it is stopped, find