DP IB Maths: AA SL

Topic Questions

3.5 Trigonometric Functions & Graphs

1a
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1 mark

The graph below shows the curve with equation y space equals space sin space 2 x in the interval negative 60 degree less or equal than x less or equal than 270 degree.

q1a-3-5-medium-ib-aa-sl-maths

Point A has coordinates left parenthesis negative 45 degree comma negative 1 right parenthesis and is the minimum point closest to the origin. Point B is the maximum point closest to the origin. State the coordinates of B.

1b
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2 marks

A straight line with equation y space equals space minus 1 half meets the graph of y equals sin space 2 x at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates open parentheses negative 15 degree comma negative 1 half close parentheses, use graph symmetries to determine the coordinates of Q and R.

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2
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4 marks
(i)
Sketch the graph of space y equals cos space left parenthesis theta plus 30 degree right parenthesis in the interval negative 180 degree less or equal than theta less or equal than 360 degree.

(ii)
Write down all the values where cos space left parenthesis theta plus 30 degree right parenthesis equals 0 in the given interval.

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3
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3 marks

A dolphin is swimming such that it is diving in and out of the water at a constant speed.

On each jump and dive the dolphin reaches a height of 2 m above sea level and a depth of 2 m below sea level.

Starting at sea level, the dolphin takes fraction numerator 2 straight pi over denominator 3 end fraction seconds to jump out of the water, dive back in and return to sea level.

Write down a model for the height, h straight m, of the dolphin, relative to sea level, at time t, in the form h equals A space sin space left parenthesis B t right parenthesis where A space and space B  are constants to be found.

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4
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6 marks

A section of a new rollercoaster has a series of rises and falls. The vertical displacement of the rollercoaster carriage, y, measured in metres relative to a fixed reference height, can be modelled using the function y equals 30 space cos space left parenthesis 24 t right parenthesis degree,  where t is the time in seconds.

Sketch the function for the interval space 0 less or equal than t less or equal than 30.

How many times will the rollercoaster carriage fall during the 30 seconds?

How long does the model suggest it will take for the rollercoaster carriage to reach the bottom of the first fall?

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5a
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3 marks

The height, h space straight m, of water in a reservoir is modelled by the function

h left parenthesis t right parenthesis equals A plus B space sin space open parentheses straight pi over 6 t close parentheses comma space t greater or equal than 0

where t is the time in hours after midnight. A and B are positive constants.

In terms of A and B, write down the natural height of the water in the reservoir, as well as its maximum and minimum heights.

5b
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3 marks

The maximum level of water is 3 m higher than its natural level.

The level of water is three times higher at its maximum than at its minimum.

Find the maximum, minimum and natural water levels.

5c
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3 marks
(i)
How many times per day does the water reach its maximum level?
(ii)
Find the times of day when the water level is at its minimum?

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6a
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2 marks

A lifejacket falls over the side of a boat from a height of 3 space straight m.
The height, h space straight m, of the lifejacket above or below sea level open parentheses h equals 0 close parentheses, at time t seconds after falling, is modelled by the equation  h equals 3 e to the power of negative 0.7 t end exponent space cos space 4 t.

The lifejacket reaches its furthest point below sea level after 0.742 seconds.
Find the total distance it has fallen, giving your answer to three significant figures.

6b
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2 marks

Write down the value of t for the first three times the lifejacket is at sea level.

6c
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3 marks
(i)
Find the value of 3 e to the power of negative 0.7 t end exponent when t equals 6.2.

(ii)
Hence justify why, from 6.2 seconds on, the lifejacket will always be within 4 centimetres of sea level.

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7a
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5 marks

The number of daylight hours, h, in the UK, during a day d days after the spring equinox (the day in spring when the number of daylight hours is 12), is modelled using the function

h equals 12 plus 9 over 2 space sin space open parentheses fraction numerator 2 pi over denominator 365 end fraction space d close parentheses

(i)
Find the number of daylight hours during the day that is 100 days after the spring equinox.
(ii)
Find the number of days after the spring equinox that the two days occur during which the number of daylight hours is closest to 9.
7b
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3 marks

For how many days of the year does the model suggest that the number of daylight hours exceeds 15 hours?  Give your answer as a whole number of days.

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8a
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3 marks

Felicity is a keen ice skater and has entered a competition that requires her to skate in a circular pathway in front of three judges. Her distance, d meters, away from the judges table, t spaceseconds after commencing her routine can be modelled by the function

d equals 12 space cos space pi over 30 t space plus 15

(i)
State the distance Felicity is away from the judges table at the start of her                 
(ii)
State the distance Felicity is away from the judges table after 15 seconds.
8b
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2 marks

Find, in terms of pi, the circumference of Felicity’s circular pathway on the ice rink.

8c
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2 marks

Find, in terms of pi, Felicity’s average speed for each lap on the ice rink.

8d
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3 marks

Felicity’s routine took three laps in total around the ice rink.

Find the times during Felicity’s routine where she was at a distance of 21 spacemetres from the judges table.

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