91Ó°ÊÓ

The data on the next page represent the average monthly temperatures for Washington, D.C. (a) Draw a scatter plot of the data for one period. (b) Find a sinusoidal function of the form \(y=A \sin (\omega x-\phi)+B\) that models the data. $$ \begin{array}{|lc|} \hline \text { Decade, } x & \text { Major Hurricanes, } \boldsymbol{H} \\ \hline 1921-1930,1 & 17 \\ 1931-1940,2 & 16 \\ 1941-1950,3 & 29 \\ 1951-1960,4 & 33 \\ 1961-1970,5 & 27 \\ 1971-1980,6 & 16 \\ 1981-1990,7 & 16 \\ 1991-2000,8 & 27 \\ 2001-2010,9 & 33 \\ \hline \end{array} $$ (c) Draw the sinusoidal function found in part (b) on the scatter plot. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on a scatter plot of the data. $$ \begin{array}{lc|} \hline \text { Month, } \boldsymbol{x} & \begin{array}{c} \text { Average Monthly } \\ \text { Temperature, }^{\circ} \mathrm{F} \end{array} \\ \hline \text { January, } 1 & 36.0 \\ \text { February, } 2 & 39.0 \\ \text { March, } 3 & 46.8 \\ \text { April, } 4 & 56.8 \\ \text { May, } 5 & 66.0 \\ \text { June, } 6 & 75.2 \\ \text { July, } 7 & 79.8 \\ \text { August, } 8 & 78.1 \\ \text { September, } 9 & 71.0 \\ \text { October, } 10 & 59.5 \\ \text { November, } 11 & 49.6 \\ \text { December, } 12 & 39.7 \end{array} $$

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \sec 540^{\circ} $$

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\cos \theta=\frac{\sqrt{2}}{2}$$

The following data represent the average monthly temperatures for Baltimore, Maryland. (a) Draw a scatter plot of the data for one period. (b) Find a sinusoidal function of the form \(y=A \sin (\omega x-\phi)+B\) that models the data.\ (c) Draw the sinusoidal function found in part (b) on the scatter plot. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on a scatter plot of the data. $$ \begin{array}{lc} \hline & \text { Average Monthly } \\ \text { Month, } \boldsymbol{x} & \text { Temperature, }^{\circ} \mathrm{F} \\\ \hline \text { January, } 1 & 32.9 \\ \text { February, } 2 & 35.8 \\ \text { March, } 3 & 43.6 \\ \text { April, } 4 & 53.7 \\ \text { May, } 5 & 62.9 \\ \text { June, } 6 & 72.4 \\ \text { July, } 7 & 77.0 \\ \text { August, } 8 & 75.1 \\ \text { September, } 9 & 67.8 \\ \text { October, } 10 & 56.1 \\ \text { November, } 11 & 46.5 \\ \text { December, } 12 & 36.7 \end{array} $$

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \cot 390^{\circ} $$

Convert each angle in degrees to radians. Express your answer as a multiple of \(\pi .\) \(540^{\circ}\)

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \sec 420^{\circ} $$

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, April 21,2018 , in Sitka, Alaska, high tide occurred at 4: 51 AM (4.85 hours) and low tide occurred at 11:50 AM (11.83 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 10.03 feet, and the height of the water at low tide was -0.46 feet. (a) Approximately when did the next high tide occur? (b) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (c) Use the function found in part (b) to predict the height of the water at 3 PM.

According to the Old Farmer's Almanac, in Miami, Florida, the number of hours of sunlight on the summer solstice of 2018 was \(13.75,\) and the number of hours of sunlight on the winter solstice was 10.52 . (a) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1 , the 91 st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer's Almanac, and compare the actual hours of daylight to the results found in part (b).

According to the Old Farmer's Almanac, in Detroit, Michigan, the number of hours of sunlight on the summer solstice of 2018 was \(15.27,\) and the number of hours of sunlight on the winter solstice was 9.07 . (a) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April \(1,\) the 91 st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer's Almanac, and compare the actual hours of daylight to the results found in part (b).