Tracy phase shift by the information given
Tracy Gitonga Modeling Data With Trigonometric Functions Precalculus 1113-213 October 18, 2011 Real-life math is used in many activities that people do in a daily basis. In the next few paragraphs I will be explaining how to use a real world data and model it with a sine function of the form of y= a sin K (x-b). The graphs will use the law of sine which is defines as, “a law stating that the ratio of the sine of an arc of a spherical triangle to the sine of the opposite angle is the same for all three arcs. The following table gives the number of hours of daylight in Philadelphia, Pennsylvania. Day | | Mar 21 | Apr 21 | May 21 | | June 21 | | July 21 | | Aug 21 | | Sept 21 | | Oct 21 | | Nov 21 | | Dec 21 | Hours of Day-light | | 12 | 13. 7 | 14.
2 | | 14. 8 | | 14. 2 | | 13.
7 | | 12 | | 11. 4| | 9. 8 | | 9. 2 | While plotting the data, I realized that the sine function works better because the graph begins from the point (0) on the x-axis of the graph.
Firstly, I realized I need to be able to find my equation before formulating a graph.My equation is y= a sin K (x-b). I need to find the “a”, which stands for amplitude, “the maximum extent of a vibration or oscillation, measured from the position of equilibrium. ” I took the difference between the highest and lowest daylight hours reading in the table and divided by two. 14. 8 – 9. 2 = 5.
6/2 A= 2. 8 Secondly, I figured the period which is defined as, “a space of time between two events or a portion of time. ” The formula is 2 /K 2 /k = 365 k = 2 /365 P= 2 /365 Thirdly, the phase shift “represents the amount a wave has shifted orizontally from the original wave. ” I was able to figure the phase shift by the information given which stated that, “the time to be in days. March 21 is the 80th day of the year. ” With that given I just took the number before what was given equaling it to 79. Our phase shift stands for b in our formula.
B = 79 Fourthly, I was able to determine the vertical shift for the function which is defined as “the vertical displacement of a function above or below the horizontal axis,” by looking at my information given and realizing that the graph begins will increase by 12.VS= +12 I now know my Start and End of the graph. Start: 79 End: 79+365=444 Therefore, after finding my entire variables, I am able to figure out my equation which is: Y = 2.
8 Sin 2 /365 (X-79)| The graph below demonstrates the above equation. Using my equation y= 2. 8 Sin 2 /365 (x-79), I will estimate the number of hours of daylight on April 1 and I will determine when there would be ten hours of daylight that year. I calculated the amount of days from January 1st to March 31st which equals to 90 days. I then plugged it in my equation in the calculator and came up with . 27 to round it of two the tenth would equal to . 5 I then added 12 because of the number of days given.
The number of hours of daylight on April 1st = 12. 5 To determine when there will be ten hours of daylight in that year, I first calculated the number of days from January to October 31. This equaled to 304 days. However, when I put that number in the calculator I would get an odd number so I went ahead and added 4 to the 304 to equal 308 because I need to get a solid number to get a particular date which is November 5th as the first date.
For the second date I took the 304 which I had previously added and added November through January dates to equal to 396. However, that number as well on the graph did not equal any specific date so I went ahead and added 2 to equal 398 and that date came to February 3rd. There would be 10 hours of daylight that year on February 3rd and November 5th.
Reference “Definition of Sine, Cosine, Tangent | Cramster. com. ” Get Homework Help in Math, Algebra, Physics, Chemistry, Science, History, Accounting, English | Cramster. com.
Chegg, June 2008. Web. 18 Oct.
2011. <http://www. cramster.