Chapter 8

Convert each of the following degree measurements to radians: a. \(50^{\circ}\) b. \(120^{\circ}\) c. \(-15^{\circ}\)

Convert each of the following radian measurements to degrees: a. \(0.25\) radian b. 1 radian c. \(-1.5\) radians

Find \(\tan \theta\) if \(\sin \theta=\frac{4}{5}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Find \(\csc \theta\) if \(\cot \theta=\frac{\sqrt{5}}{2}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Starting with the addition formulas for the sine and cosine, derive these identities: \(\cos \left(\frac{\pi}{2}+\theta\right)=-\sin \theta \quad\) and \(\quad \sin \left(\frac{\pi}{2}+\theta\right)=\cos \theta\) Give geometric arguments to justify the identities.

Use the addition formulas for sine and cosine to derive the double-angle formulas $$ \begin{aligned} \sin (2 A) &=2 \sin A \cos A \\ \cos (2 A) &=\cos ^{2} A-\sin ^{2} A \\ &=2 \cos ^{2} A-1 \\ &=1-2 \sin ^{2} A \end{aligned} $$

a. Use the double-angle formulas along with the Pythagorean identity \(\sin ^{2} A+\cos ^{2} A=1\) to show that \(\cos ^{2} \theta=\frac{1}{2}(1+\cos 2 \theta) \quad\) and \(\sin ^{2} \theta=\frac{1}{2}(1-\cos 2 \theta)\) b. Use the identities in part (a) to show that $$ \begin{aligned} &\int \cos ^{2} x d x=\frac{1}{2} x+\frac{1}{4} \sin (2 x)+C \\ &\int \sin ^{2} x d x=\frac{1}{2} x-\frac{1}{4} \sin (2 x)+C \end{aligned} $$ and c. An object moves along a straight line in such a way that after \(t\) seconds, its velocity is given by $$ v(t)=2 t+\sin ^{2}\left(\frac{\pi t}{6}\right) $$ meters per second. Find the average velocity of the object over the time period \(0 \leq t \leq 3\).

Differentiate the given function. $$f(x)=\cos (1-5 x)$$

Differentiate the given function. $$f(x)=\sin (3 x+1) \cos x$$

Differentiate the given function. $$f(x)=\cos ^{2} x$$

Differentiate the given function. $$f(x)=\tan \left(3 x^{2}+1\right)$$

Differentiate the given function. $$f(x)=\tan ^{2}(3 x+1)$$

Differentiate the given function. $$f(x)=\frac{\sin x}{1-\cos x}$$

Differentiate the given function. $$f(x)=\ln \left(\cos ^{2} x\right)$$

Differentiate the given function. $$f(x)=e^{-2 x} \cos 3 x$$

Find the indicated integral. $$\int(\sin 2 t+\cos 2 t) d t$$

Find the indicated integral. $$\int \cos (1-2 t) d t \quad\( \)

Find the indicated integral. $$\int \sin x \cos x d x$$

Find the indicated integral. $$\int x \sin x d x$$

Find the indicated integral. $$\int \frac{\sec ^{2} t}{\tan t} d t$$

Find the indicated integral. $$\int_{0}^{\pi} \cos \left(\frac{x}{3}\right) d x$$

Find the indicated integral. $$\int_{0}^{1} x \sin \left(x^{2}\right) d x$$

In each of the following cases, use the graphing utility of your calculator to draw the graphs of the given pair of functions \(f(x)\) and \(g(x)\) on the same screen. Describe the relationship between the graphs of \(f(x)\) and \(g(x)\). a. \(f(x)=\sin x\) and \(g(x)=2 \sin x\) b. \(f(x)=\cos x\) and \(g(x)=2 \cos 2 x\) c. \(f(x)=\sin x\) and \(g(x)=\sin \left(x+\frac{\pi}{2}\right)\) d. \(f(x)=\cos x\) and \(g(x)=2+\cos x\)

Use your calculator to solve the equation \(2 \tan 3 x-5.87=2 \sin 2 x \quad\) for \(0 \leq x \leq \frac{\pi}{2}\) to three decimal places.

Find the area of the region bounded by the curves \(y=\sin 2 x\) and \(y=\cos x\) over the interval \(\frac{\pi}{6} \leq x \leq \frac{\pi}{2}\)

Let \(R\) be the region bounded by the \(x\) axis, the curve \(y=\cos x+\sin x\), and the lines \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{6}\). Find the volume of the solid generated by rotating \(R\) about the \(x\) axis.

a. Find the period \(p\), the amplitude \(b\), the horizontal shift \(d\), and the vertical shift \(a\) of the function \(f(x)=5.0+3.0 \cos \left[\frac{\pi}{4}(x-1.5)\right]\) b. Sketch the graph of the function \(f(x)\) in part (a).

a. Find the period \(p\), the amplitude \(b\), the horizontal shift \(d\), and the vertical shift \(a\) of the function \(f(x)=33+27 \cos \left[\frac{2 \pi}{25}(x-11)\right]\) b. Sketch the graph of the function \(f(x)\) in part (a).

The maximum daily temperature \(T(x)\) in degrees Celsius in Minneapolis on day \(x\) of the year can be modeled as $$ T(x)=13+33 \cos \left[\frac{2 \pi}{365}(x-271)\right] $$ where \(x=0\) corresponds to January 1 . a. Using a calculator, find the maximum daily temperature in Minneapolis on the first day of January. Repeat for the first days of March, May, July, September, and November. b. Find the largest and smallest maximum daily temperature in Minneapolis during the year. c. Draw the graph of the maximum daily temperature function \(T(x)\).

The ozone levels in parts per million (ppm) in a city can be modeled by the function \(F(t)=0.01 t^{3}+0.05 t^{2}+1.1 t+56+22 \sin (2 \pi t)\) where \(t\) is the time in years after 1990 . a. Find the levels of ozone on July 1,1990 . Repeat for January 1, 2000, and March 1, \(2005 .\) b. Find the rate of change of the level of ozone on the three dates in part (a). c. Graph \(F(t)\) for the time period from 1990 to \(2010(0 \leq t \leq 20)\). d. Describe the behavior of \(F(t)\) as \(t\) increases from 0 to 20 . Interpret the roles of the polynomial part of \(F(t)\) and the periodic part.

The number of hours of daylight in New York for day \(t\) of the year can be modeled by the function $$ D(t)=12.2+3.09 \cos \left[\frac{2 \pi}{365}(t-185)\right] $$ where \(t=0\) corresponds to January 1 . a. How many hours of daylight are there on January 1? On March \(15(t=74)\) ? On June 21 \((t=172)\) ? b. On which day of the year is the number of daylight hours the greatest? When does the least number of daylight hours occur? c. What is the average number of daylight hours per day over the entire year \((0 \leq t \leq 365) ?\)

On New Year's Eve, Zain is watching the descent of a lighted ball from atop a tall building that is 600 feet away. The ball is falling at the rate of 20 feet per minute. At what rate is the angle of elevation of Zain's line of sight changing with respect to time when the ball is 800 feet from the ground?

Solve the separable differential equation $$ \frac{d y}{d x}=\sin x \sec y $$ subject to the condition \(y=1\) when \(x=0\).

Use the graphing utility of your calculator to draw the curves \(y=\sin x\) and \(y=e^{x-2}\) for \(x \geq 0\) on the same screen. Find all points of intersection of the two curves. Let \(R\) be the region enclosed by the two curves. a. Find the area of the region \(R\). b. Find the volume of the solid formed by revolving the region \(R\) around the \(x\) axis. \([\) Hint: It may help to recall the identity \(\sin ^{2} x=\frac{1-\cos 2 x}{2}\).] c. Check the integration in part (b) by using the numeric integration feature of your calculator.

Deal with topics developed in Chapter \(7 .\) What is the largest possible value of the product \(f(A, B, C)=\sin A \sin B \sin C\) given that \(A, B\), and \(C\) are the angles in a triangle? [Hint: It may help to note that \(A+B+C=\pi\).]

Deal with topics developed in Chapter \(7 .\) POLARIZED LIGHT A polarized light wave travels in such a way that its vertical displacement \(y\) at time \(t\) is a function of both \(t\) and its horizontal displacement \(x\) according to the formula $$ y(x, t)=0.27 \sin \left(10 \pi t-3 \pi x+\frac{\pi}{4}\right) $$ a. Find \(\frac{\partial y}{\partial x}\) and \(\frac{\partial y}{\partial t}\). b. For what points \((x, t)\) is \(y(x, t)\) maximized? For what points is \(y(x, t)\) minimized?

Deal with topics developed in Chapter \(7 .\) The following equation involving partial derivatives of the function \(u(x, t)\) is called the diffusion equation: $$ u_{t}=c^{2} u_{x x} $$ The diffusion equation is used in modeling a large variety of physical phenomena. For instance, in biology it is used to model the mechanism for butterfly wing patterns, the effects of genetic drift, and macrophage response to bacteria in the lungs, while in physics, it is used to study the motion of molecules and heat conduction. a. Show that the function \(u=e^{-c^{2} k^{2} t} \sin k x\) satisfies the diffusion equation. b. Read an article on the diffusion equation, and write a paragraph on one of its applications.

Deal with topics developed in Chapter \(7 .\) Ground temperature models are important in ecology, where they are used to study phenomena such as frost penetration. Suppose ground temperature \(T\) at time \(t\) (months) and depth \(x\) (centimeters) is modeled by a function of the form $$ T(x, t)=A+B e^{-k x} \sin (a t-k x) $$ where \(a=\frac{\pi}{6}\) and \(A, B\), and \(k\) are positive constants. a. Find the partial derivatives \(T_{x}\) and \(T_{r}\). b. The partial derivative \(T_{x}\) measures the rate at which the ground temperature drops with increasing depth for fixed time. Give a similar interpretation for the partial derivative \(T_{r-}\) c. Show that \(T(x, t)\) satisfies the diffusion equation \(T_{r}=c^{2} T_{x=}\) where \(c\) is a constant involving \(B\) and \(k\).

Deal with topics developed in Chapter \(7 .\) Find the largest and the smallest values of the function $$ f(x, y)=2 \sin x+5 \cos y $$ over the rectangle \(R\) with vertices \((0,0),(2,0)\), \((2,5)\), and \((0,5)\).