Hong Kong
Stage 4 - Stage 5

# Applications of sine and cosine functions including phase shift (radians)

## Interactive practice questions

The number of hours of daylight at a point on the Antarctic Circle is given approximately by $L=12+12\cos\left(\frac{2\pi}{365}\left(t+10\right)\right)$L=12+12cos(2π365(t+10)), where $t$t is the number of days that have elapsed since 1 January.

a

Find $L$L on 22 June, the winter solstice, which is $172$172 days after 1 January. Round your answer to five decimal places.

b

Write your answer to part (a) in seconds, rounding to two decimal places.

c

Find $L$L on 21 March, the vernal equinox, which is $79$79 days after 1 January. Round your answer to two decimal places.

d

How many whole days will there be until the first time there is less than $5$5 hours of daylight?

Easy
11min

Tobias is jumping on a trampoline. Victoria watches him bounce at a regular rate and wants to try to model his height over time. When Victoria starts her stopwatch, Tobias is at a minimum height of $30$30 cm below the trampoline frame. A moment later Victoria records Tobias reaching a maximum height of $50$50 cm above the trampoline frame. She uses the function $H\left(s\right)=a\sin\left(2\pi\left(s-c\right)\right)+d$H(s)=asin(2π(sc))+d, where $H$H is the height in cm above the trampoline frame and $s$s is the time in seconds.

Easy
13min

The water level on a beach wall is given by $h\left(t\right)=6+4\cos\left(\frac{\pi}{6}t-\frac{\pi}{3}\right)$h(t)=6+4cos(π6tπ3), where $t$t is the number of hours after midnight and $h$h is the depth of the water in metres.

Medium
8min

The height in metres of the tide above mean sea level is given by $h=4\sin\left(\frac{\pi\left(t-2\right)}{6}\right)$h=4sin(π(t2)6), where $t$t is the time in hours since midnight.

Hard
13min