Investing Trig Graphs Project: A Trip to the Beach Culminating Task On Amy’s recent visit to the beach during spring break she realized she could not get away from the math she was learning in Trigonometry. Everywhere she looked she saw periodic functions. It all started when her dad wanted to know when the sun would rise in the morning. When she glanced at his sunrise/sunset chart Amy thought the data looked very periodic. st Sunrise/Sunset Times on the 1 of each month last year Times are given in a 24 hour format.
Jan
Feb April
March May Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset 725
1735
718
1802
652 1824 615 1845 541 1905 523 1925 July
Aug Oct
Sept Nov Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset Sunrise
Sunset 526 1934 543 1921 602 1849 620 1811 642 1737 707 1723 1. Graph the data using your calculator and see if you agree that the data looks periodic.
(Graph sunset separately from sunset.)
2. Use your calculator to find a trigonometric regression equation for the sunrise and
sunset times (let t=1 represent January).
Dec Sunrise
June 3. Test a few values of t and discuss the accuracy of your regression equation.
4. Explain the values your calculator reported for a, b, c, and d in terms of the data and
the graph.
Later that day, Amy was building a s and castle with her brother, Robert. Just as they were putting the finishing touches on it, the tide came in and washed it all away. 5. Write a sine function which models the tide if the equilibrium point is 4.26 feet and
the highest point the tide reaches is at 10.6 feet. The phase shift is -3.5 hours and the
period is 12 .6 hours.
6. Using your equation determine when the tide will be at 8 feet.
When Amy went swimming with her mother, Nancy, they both saw a buoy floating in the water. As Amy lay on her float she watched the way the buoy bobbed up and down in the water and remembered a problem she had worked on in class. It all came back to her like a bad dream. The Signal Buoy
During a summer squall a signal buoy moved a total distance of 4 feet from its lowest point to its highest point. It returned to its highest point every 25 seconds. It was at its highest point at t=0. 7. Complete the problem Amy worked on in her class. First, find an equation that
models the motion of the buoy.
8. Assuming the motion of the storm waves stays consistent for an hour, determine the
height of the buoy at 5 seconds and at 5 minutes.
9. After the storm passed, the total distance moved from the highest point to the lowest
point was only 1.5 feet. The buoy returned to its highest point after 40 seconds. How
does this change your equation from a?
10. At what time was the buoy 1.5 feet above the equilibrium point?
Later that night Amy’s family went to the amusement park. She decides to ride the ferris wheel so she can look out at the ocean. She was disappointed to find out that a 4 5 foot building blocked her view for part of the ride. Amy’s height from the ground as she travels around the Ferris wheel can be found using the following equation where t = time in seconds from the beginning of Amy’s ride. ℎ = 30.48 sin
2n
t + 34.27
4.2S
11. How long will it take until Amy can see over the building?
12. How long will Amy be able to see the ocean?
13. Amy rides for 17 minutes. If she first started riding at 8:10 pm, at what times will the
ocean come into view?
On the way back home, Amy and her brother started arguing about the sin and cos functions. Robert says the graphs of the two functions are unrelated to each other. Amy says you can take any sin function and write it in the form of a c osine function. 14. Who do you think is correct? Write a convincing argument stating why that person is
correct.
A m y finally makes it h ome and runs to her room to get away from her annoying brother. Right before she falls asleep her friend, Sydney, calls to remind her about the s ome make-­‐-­‐-­‐up work she has to turn in tomorrow. She quickly jumps out of bed and looks at her problems with dismay. Check Amy’s answers to the problems while she gets some much needed rest. Mark the problems as correct or incorrect. Correct any problems she missed. (Her answers are circled.) 15. cos (tan
-1
1) =
3√2
17.
sin (sin
-1 √3 2 )=
2 √3 2 16. Arctan (-1) =
rr
4 18. sin(arctan √3 + arcsin ½ ) = 2