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.

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
Approx 19 minutes

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.

A circular Ferris wheel that is $40$40 meters in diameter contains several carriages. Hannah and Michael enter a carriage at the bottom of the Ferris wheel, and get off $6$6 minutes later after having gone around completely $3$3 times. When a carriage is at the bottom of the wheel, it is $1$1 meter above the ground.

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.

### Outcomes

#### M8-2

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