Trig Functions and Graphs (radians)

Hong Kong

Stage 4 - Stage 5

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+12`c``o``s`(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`)=`a``s``i``n`(2π(`s`−`c`))+`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+4`c``o``s`(π6`t`−π3), 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`=4`s``i``n`(π(`t`−2)6), where $t$`t` is the time in hours since midnight.

Hard

13min

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