91Ó°ÊÓ

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \csc \frac{2 \pi}{3} $$

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sec 42^{\circ} 12^{\prime}\) (b) \(\csc 48^{\circ} 7^{\prime}\)

Find a model for simple harmonic motion satisfying the specified conditions. \(\begin{array}{ll}\text { Displacement } (t=0) & \text { Amplitude }\end{array}\) Period $$ \begin{array}{lll} 0 & 3 \text { meters } & 6 \text { seconds } \end{array} $$

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \sec 1.8 $$

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t(t>0),\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium \((y=0)\).

NUMERICAL AND GRAPHICAL ANALYSIS The cross section of an irrigation canal is an isosceles trapezoid of which 3 of the sides are 8 feet long (see figure). The objective is to find the angle \(\theta\) that maximizes the area of the cross section. [Hint: The area of a trapezoid is \((h / 2)\left(b_{1}+b_{2}\right)\).] (a) Complete seven additional rows of the table. $$ \begin{array}{|c|c|c|c|} \hline \text { Base } 1 & \text { Base } 2 & \text { Altitude } & \text { Area } \\ \hline 8 & 8+16 \cos 10^{\circ} & 8 \sin 10^{\circ} & 22.1 \\ \hline 8 & 8+16 \cos 20^{\circ} & 8 \sin 20^{\circ} & 42.5 \\ \hline \end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the maximum cross-sectional area. (c) Write the area \(A\) as a function of \(\theta\). (d) Use a graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that of part (b)?

The table shows the average sales \(S\) (in millions of dollars) of an outerwear manufacturer for each month \(t,\) where \(t=1\) represents January. $$ \begin{aligned} &\begin{array}{|c|c|c|c|c|c|c|} \hline \text { Time, } t & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Sales, } S & 13.46 & 11.15 & 8.00 & 4.85 & 2.54 & 1.70 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time, } t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Sales, } S & 2.54 & 4.85 & 8.00 & 11.15 & 13.46 & 14.30 \\ \hline \end{array} \end{aligned} $$ (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model's amplitude in the context of the problem.

You are standing 45 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86 th floor (the observatory) is \(82^{\circ} .\) If the total height of the building is another 123 meters above the 86 th floor, what is the approximate height of the building? One of your friends is on the 86 th floor. What is the distance between you and your friend?

Convert the angle measure from degrees to radians. Round to three decimal places. $$ -48.27^{\circ} $$

Use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. $$ y=\frac{1}{100} \sin 120 \pi t $$