Trig Functions and Graphs (radians)

Hong Kong

Stage 4 - Stage 5

The population (in thousands) of two different types of insects on an island can be modelled by the following functions: Butterflies: $f\left(t\right)=a+b\sin\left(mt\right)$`f`(`t`)=`a`+`b``s``i``n`(`m``t`), Crickets: $g\left(t\right)=c-d\sin\left(kt\right)$`g`(`t`)=`c`−`d``s``i``n`(`k``t`)

$t$`t` is the number of years from when the populations started being measured, and $a$`a`,$b$`b`,$c$`c`,$d$`d`,$m$`m`, and $k$`k` are positive constants. The graphs of $f$`f` and $g$`g` for the first $2$2 years are shown below.

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a

State the function $f\left(t\right)$`f`(`t`) that models the population of Butterflies over $t$`t` years.

b

State the function $g\left(t\right)$`g`(`t`) that models the population of Crickets over $t$`t` years.

c

How many times over a $18$18 year period will the population of Crickets reach its maximum value?

d

How many years after the population of Crickets first starts to increase, does it reach the same population as the Butterflies?

e

Solve for $t$`t`, the number of years it takes for the population of Butterflies to first reach $200000$200000.

Easy

18min

Three objects, $X$`X`, $Y$`Y` and $Z$`Z` are placed in a magnetic field such that object $X$`X` is $2$2 cm from object $Y$`Y` and $4$4 cm from object $Z$`Z`. As object $X$`X` is moved closer to line $YZ$`Y``Z`, object $Y$`Y` and $Z$`Z` move in such a way that the lengths $XY$`X``Y` and $XZ$`X``Z` remain fixed.

Let $\theta$`θ` be the angle between sides $XY$`X``Y` and $XZ$`X``Z`, and let the area of triangle $XYZ$`X``Y``Z` be represented by $A$`A`.

Easy

5min

A metronome is a device used to help keep the beat consistent when playing a musical instrument. It swings back and forth between its end points, just like a pendulum.

For a particular speed, the given graph represents the metronome's distance, $x$`x`cm, from the centre of its swing, $t$`t` seconds after it starts swinging. Negative values of $x$`x` represent swinging to the left, and positive values of $x$`x` represent swinging to the right of the centre.

Medium

4min

Sounds around us create pressure waves. Our ears interpret the amplitude and frequency of these waves to make sense of the sounds.

A speaker is set to create a single tone, and the graph below shows how the pressure intensity ($I$`I`) of the tone, relative to atmospheric pressure, changes over $t$`t` seconds.

Medium

4min

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