91Ó°ÊÓ

Sketch a reasonable graph that models the given situation. The motion of a diving board vibrating 10 inches in each direction per second just after someone has dived off.

Without drawing a graph, describe the behavior of the basic tangent curve.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The bearing from \(O\) to \(B\) is \(E 70^{\circ} \mathrm{S}\).

Music and mathematics have been linked over the centuries. Group members should research and present a seminar to the class on music and mathematics. Be sure to include the role of trigonometric functions in the music- mathematics link.

Describe what happens to the tangent of an acute angle as the angle gets close to \(90^{\circ} .\)

The number of hours of daylight, \(H,\) on day \(t\) of any given year (on January \(1, t=1\) ) in Fairbanks, Alaska, can be modeled by the function $$H(t)=12+8.3 \sin \left[\frac{2 \pi}{365}(t-80)\right]$$ a. March \(21,\) the 80 th day of the year, is the spring equinox. Find the number of hours of daylight in Fairbanks on this day. b. June \(21,\) the 172 nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in Fairbanks on this day. c. December \(21,\) the 355 th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in Fairbanks on this day.

People who believe in biorhythms claim that there are three cycles that rule our behavior-the physical, emotional, and mental. Each is a sine function of a certain period. The function for our emotional fluctuations is $$E=\sin \frac{\pi}{14} t$$ where \(t\) is measured in days starting at birth. Emotional fluctuations, \(E,\) are measured from \(-1\) to \(1,\) inclusive, with 1 representing peak emotional well-being, \(-1\) representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low. a. Find \(E\) corresponding to \(t=7,14,21,28,\) and 35. Describe what you observe. b. What is the period of the emotional cycle?

Rounded to the nearest hour, Los Angeles averages 14 hours of daylight in June, 10 hours in December, and 12 hours in March and September. Let \(x\) represent the number of months after June and let \(y\) represent the number of hours of daylight in month \(x .\) Make a graph that displays the information from June of one year to June of the following year.

The height of the water, \(H,\) in feet, at a boat dock \(t\) hours after 6 A.M. is given by $$H=10+4 \sin \frac{\pi}{6} t$$ a. Find the height of the water at the dock at 6 A.M., 9 A.M., noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?

A clock with an hour hand that is 15 inches long is hanging on a wall. At noon, the distance between the tip of the hour hand and the ceiling is 23 inches. At 3 p.M., the distance is 38 inches; at 6 p.m., 53 inches; at 9 p....., 38 inches; and at midnight the distance is again 23 inches. If \(y\) represents the distance between the tip of the hour hand and the ceiling \(x\) hours after noon, make a graph that displays the information for \(0 \leq x \leq 24\).