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Chapter 4: Problem 44
Find the exact value of each expression. Do not use a calculator. $$\frac{1}{\cot \frac{\pi}{4}}-\frac{2}{\csc \frac{\pi}{6}}$$
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This exercise is intended to provide some fun with biorhythms, regardless of whether you believe they have any validity. We will use each member's chart to determine biorhythmic compatibility. Before meeting, each group member should go online and obtain his or her biorhythm chart. The date of the group meeting is the date on which your chart should begin. Include 12 months in the plot. At the meeting, compare differences and similarities among the intellectual sinusoidal curves. Using these comparisons, each person should find the one other person with whom he or she would be most intellectually compatible.
In Chapter \(5,\) we will prove the following identities: $$ \begin{aligned} \sin ^{2} x &=\frac{1}{2}-\frac{1}{2} \cos 2 x \\ \cos ^{2} x &=\frac{1}{2}+\frac{1}{2} \cos 2 x \end{aligned} $$ Use these identities to solve. Use the identity for \(\cos ^{2} x\) to graph one period of \(y=\cos ^{2} x\)
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 following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.
The number of hours of daylight in Boston is given by $$ y=3 \sin \frac{2 \pi}{365}(x-79)+12 $$ where \(x\) is the number of days after January 1 a. What is the amplitude of this function? b. What is the period of this function? c. How many hours of daylight are there on the longest day of the year? d. How many hours of daylight are there on the shortest day of the year? e. Graph the function for one period, starting on January 1
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