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New Zealand
Level 8 - NCEA Level 3

Applications of sine and cosine functions

Interactive practice questions

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+bsin(mt), Crickets: $g\left(t\right)=c-d\sin\left(kt\right)$g(t)=cdsin(kt)

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


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


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


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


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

Approx 19 minutes
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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 $YZ$YZ, object $Y$Y and $Z$Z move in such a way that the lengths $XY$XY and $XZ$XZ remain fixed.

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

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The tide rises and falls in a periodic manner, and Brad needs to chart tide levels to determine when he can sail his ship into a bay. He begins measuring at low tide at $8$8am, and sees the marking that the bay is $7$7 metres deep. He measures high tide at $2$2pm, when the bay is $15$15metres deep.



Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions

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