Lesson Worksheet: Modeling with Trigonometric Functions Mathematics

In this worksheet, we will practice modeling real-world situations using periodic functions.


Which of the following is the best model for the temperature fluctuations on a cold winter’s day in a location where the warmest part of the day is around 2 pm and the coldest around 2 am? Let 𝑇 be the temperature in degrees celsius and 𝑑 be the time after midnight in hours.

  • A𝑇=5ο€»πœ‹12(π‘‘βˆ’15)+3cos
  • B𝑇=3ο€»πœ‹12(𝑑+7)+30sin
  • C𝑇=10ο€»πœ‹6(π‘‘βˆ’8)+9sin
  • D𝑇=4ο€»πœ‹6(𝑑+13)+5sin
  • E𝑇=3ο€»πœ‹12(π‘‘βˆ’14)+2cos


London is in the northern hemisphere and the number of hours of daylight varies throughout the year. On December 21st, they have around 7 hours and 49 minutes of daylight, while on June 21st, they have 16 hours and 38 minutes. Which of the following models would best approximate the number of hours of daylight in London on a given day, where β„Ž is the number of hours of daylight and 𝑑 is the number of days since the first of January?

  • Aβ„Ž=3βˆ’5ο€Ό2πœ‹365(𝑑+11)cos
  • Bβ„Ž=12βˆ’4.4ο€Ό2πœ‹365(𝑑+11)cos
  • Cβ„Ž=24βˆ’3ο€Ό2πœ‹365(π‘‘βˆ’7)sin
  • Dβ„Ž=9βˆ’9ο€Ό4πœ‹365(π‘‘βˆ’30)sin
  • Eβ„Ž=5βˆ’4ο€Ό4πœ‹365(π‘‘βˆ’13)sin


Anthony and his friends went on a trip to the London Eye. The whole trip lasted for 30 minutes. When they got into the pod, they were 15 m above the Thames. Given that the diameter of the London Eye is 120 m, write the equation for the height, β„Ž, of the pod above the Thames 𝑑 minutes after they got into it.

  • Aβ„Ž=75βˆ’60ο€Όπœ‹π‘‘15cos
  • Bβ„Ž=75βˆ’60ο€Όπœ‹π‘‘30cos
  • Cβ„Ž=75+60ο€Όπœ‹π‘‘15cos
  • Dβ„Ž=15+ο€Όπœ‹π‘‘15cos
  • Eβ„Ž=15βˆ’ο€Όπœ‹π‘‘15cos


The depth of the water in a fishing port, 𝑆, is affected by the tides. It can be represented by 𝑆=4(15𝑛)+28sin∘, where 𝑆 is measured in meters and 𝑛 is the time elapsed, in hours, after midnight. How many times a day is the depth of the water exactly 24 meters?

  • AFour times
  • BOne time
  • CTwo times
  • DThree times


The temperature fluctuation in London over the course of a day can be modeled using a sinusoidal function.

Given that the maximum temperature of 21∘C was at 3 pm, and the minimum temperature of 10∘C was at 3 am, write an expression for the temperature in terms of 𝑑, the number of hours after midnight.

  • A𝑇=5.5ο€»πœ‹12(π‘‘βˆ’15)+15.5cos
  • B𝑇=5.5ο€»πœ‹12(𝑑+15)+15.5cos
  • C𝑇=ο€»πœ‹12(π‘‘βˆ’15)+21cos
  • D𝑇=ο€»πœ‹12(𝑑+15)+21cos
  • E𝑇=5.5ο€»πœ‹12(π‘‘βˆ’15)+21cos

Hence, find the temperature at 7 pm.

  • A𝑇=21.5∘C
  • B𝑇=10.74∘C
  • C𝑇=18.25∘C
  • D𝑇=20.13∘C
  • E𝑇=23.75∘C


Michael and Hannah went swimming in the sea at 2 pm, when it was high tide. The change in the height of water with respect to the annual average is given by β„Ž=5ο€Ό4πœ‹25π‘‘οˆcos, where 𝑑 is the time, in hours, after any high tide.

At what time was the next high tide?

  • A2:30 pm
  • B12:00 am
  • C2:00 am
  • D2:00 pm
  • E2:30 am

When will it be high tide in the afternoon three days later?

  • A5 pm
  • B12 am
  • C12 pm
  • D5 am
  • E5:30 pm

They want to go back to the same beach three days later, in the afternoon, and would like the height of the water to be at least 4 meters above the annual height average. Between what times should they go?

  • ABetween 6:15 pm and 4:12 am
  • BBetween 1:17 pm and 11:12 pm
  • CBetween 3:17 pm and 1:12 am
  • DBetween 3:42 pm and 6:17 pm
  • EBetween 12:42 pm and 3:17 pm


The number of hours of daylight in Paris depends on the season, and it is modeled by 𝑑=12βˆ’4ο€Ό2πœ‹365(𝑑+10)cos, where 𝑑 is the number of the days in a year (January 1 is day 1). According to this model, on what days does daylight in Paris last 10 hours?

  • AFebruary 20, April 21
  • BJanuary 20, May 22
  • CAugust 21, October 21
  • DApril 21, August 21
  • EFebruary 20, October 21


A particle moves along the π‘₯-axis so that its displacement from the origin 𝑂 after 𝑑 seconds is 7(12𝑑)sin meters. Find the times at which the particle’s displacement is βˆ’72 meters. Use 𝑛 to denote an arbitrary nonnegative integer.

  • A𝑑=ο€»πœ‹72+π‘›πœ‹6seconds, 𝑑=ο€Ό7πœ‹6+2π‘›πœ‹οˆseconds
  • B𝑑=ο€Ό7πœ‹72+π‘›πœ‹6seconds, 𝑑=ο€Ό7πœ‹6+2π‘›πœ‹οˆseconds
  • C𝑑=ο€Ό72+π‘›πœ‹6seconds, 𝑑=ο€Ό72+2π‘›πœ‹οˆseconds
  • D𝑑=ο€»πœ‹72+π‘›πœ‹6seconds, 𝑑=ο€Ό72+2π‘›πœ‹οˆseconds
  • E𝑑=ο€Ό7πœ‹72+π‘›πœ‹6seconds, 𝑑=ο€Ό11πœ‹72+π‘›πœ‹6seconds


The depth of the water in a fishing port is usually 29 meters. The tidal movement is represented by 𝑆=3(15𝑛)+29cos∘, where 𝑛 is the time elapsed in hours after midnight. How many times in a day is the depth of the water 32 meters?

  • AThree times
  • BOne time
  • CTwo times
  • DFour times


In Portsmouth harbour, the height of the tide during December is modeled using the following function β„Ž=5ο€Ό4πœ‹π‘‘25sin where β„Ž is the height of the tide, measured in meters, above the annual average and 𝑑 is the time, in hours, after midnight on the first of the month.

During the spring, the tidal variation is 50% higher than the December variation. Write an equation for the height of the tide during April.

  • Aβ„Ž=7.5ο€Ό4πœ‹π‘‘25sin
  • Bβ„Ž=2.5ο€Ό4πœ‹π‘‘25sin
  • Cβ„Ž=2.5ο€Ό8πœ‹π‘‘25sin
  • Dβ„Ž=5ο€Ό2πœ‹π‘‘25sin
  • Eβ„Ž=5ο€Ό8πœ‹π‘‘25sin

During June, the tidal variation is the same as December. However the average height of the water is 1 m higher. Write an equation for the height of the tide during June.

  • Aβ„Ž=5ο€½4πœ‹(𝑑+1)25sin
  • Bβ„Ž=6ο€Ό4πœ‹π‘‘25sin
  • Cβ„Ž=5ο€Ό4πœ‹π‘‘25+1sin
  • Dβ„Ž=5ο€Ό4πœ‹π‘‘25+1sin
  • Eβ„Ž=5ο€Ό4πœ‹π‘‘25οˆβˆ’1sin

In January, the tidal variation and average height of the water are the same as in December. However, the first high tide of the month is 3 hours later than the first high tide in December. Write an equation for the height of the tide in January.

  • Aβ„Ž=5ο€½4πœ‹(π‘‘βˆ’3)25+3sin
  • Bβ„Ž=15ο€½4πœ‹(π‘‘βˆ’3)25sin
  • Cβ„Ž=5ο€½4πœ‹(π‘‘βˆ’3)25ο‰βˆ’3sin
  • Dβ„Ž=5ο€½4πœ‹(π‘‘βˆ’3)25sin
  • Eβ„Ž=5ο€½4πœ‹(𝑑+3)25sin


Benjamin sits on a pier, his feet dangling 60 cm below the pier. The pier is usually 80 cm above the lake. But this is a windy day, and waves make the depth of the lake oscillate. The depth of the lake under the pier is given, in meters, by 𝑑=𝑑+0.3ο€Ό2πœ‹10π‘‘οˆοŠ¦sin, where π‘‘οŠ¦ is the depth of the lake on a quiet day, and 𝑑 is the time in seconds. What fraction of the time does Benjamin have his feet under water? Express your answer as a percentage correct to one decimal place.

  • A38.4%
  • B76.8%
  • C23.2%
  • D53.5%
  • E26.8%


Sophia and her friends got on a Ferris wheel. When they entered the cabin at 3:15 pm, they were 5 m above the ground. The height of the cabin 𝑑 minutes after they got into it is given by β„Ž=25βˆ’20πœ‹π‘‘10cos. When were they 40 m above the ground? Give your answer to the nearest minute.

  • A3:23 pm, 3:17 pm
  • B3:28 pm, 3:32 pm
  • C3:23 pm, 3:27 pm
  • D3:17 pm, 3:33 pm
  • E3:27 pm, 3:33 pm


Emma is jumping on a trampoline. Her height β„Ž above the trampoline , in meters, is given by β„Ž=1βˆ’ο€»πœ‹2𝑑cos, at 𝑑 seconds after she started jumping.

How many seconds after each rebound does it take her to reach a height of 50 cm during the descent? Round your answer to the nearest tenth of a second.

  • A0.3 s
  • B3.3 s
  • C1.7 s
  • D2.6 s
  • E0.7 s

What fraction of the time is Emma at least 1.2 m above the trampoline? Express your answer as a percentage correct to one decimal place.

  • A56.4%
  • B43.6%
  • C87.2%
  • D71.8%
  • E28.2%


The outside temperature (in degrees Celsius) on a certain day was modeled with 𝑇=12+7ο€»πœ‹12(π‘‘βˆ’10)sin, where 𝑑 is the time after midnight in hours. At what times of the day was the temperature 10∘C? Give your answer to the nearest minute using a 24-hour format.

  • A11:06, 20:54
  • B8:54, 11:06
  • C20:54, 23:06
  • D8:54, 23:06
  • E17:06, 2:54


The temperature fluctuation on a cold winter’s day (in degrees Celsius) is modeled by 𝑇=3ο€»πœ‹12(π‘‘βˆ’14)+2cos, where 𝑑 is the time of the day expressed in hours after midnight. At what times of the day was the temperature 0∘C?

  • A5:13 am, 10:47 am
  • B5:13 pm, 10:47 pm
  • C5:13 am, 10:47 pm
  • D4:47 am, 11:13 am
  • E5:13 pm, 10:47 am


The daily solar irradiation 𝑄(/)Wm on a point just above Earth’s atmosphere 𝑛 days after the summer solstice is given by 𝑄(𝑛)=1,360+46ο€Ό2πœ‹365π‘›οˆcos. How many times per year is the daily solar irradiation 1,350 W/m2? How many days after the solstice does this occur?

  • ATwice, after 79 and 104 days
  • BTwice, after 104 and 261 days
  • CTwice, after 79 and 287 days
  • DTwice, after 261 and 287 days
  • ETwice, after 196 and 352 days


Chloe and Mason are at home. They see a bucket suspended on an elastic string going up and down in front of their window. They record the times at which they saw the bucket appear, disappear, appear again, and so on, as shown in the table.

Time (s)01341011
Position of the Bucket top of the windowbottom of the window bottom of the windowtop of the window top of the window bottom of the window

What is the period 𝑇 of the bucket’s movement?

The vertical displacement of the bucket with respect to the center of its movement can be modeled with the function β„Ž=β„Žο€Ό2πœ‹π‘‡π‘‘+3πœ‹5cos, where 𝑇 is the period above and 𝑑, in seconds, is measured as in the table. Given that the height of Chloe and Mason’s window is 1.2 m, what is the amplitude β„ŽοŠ¦ of the function? Give your answer to one decimal place.

At what value of 𝑑 is the bucket at its highest point?

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