Chapter 11

Find the area of each triangle. \(A=42.5^{\circ}, b=13.6\) meters, \(c=10.1\) meters

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=2+\sin \theta, r=2+\cos \theta ; 0 \leq \theta<2 \pi$$

The U.S. flag includes the colors red, white, and blue. Which color, red or white, is predominant? (Only \(18.73 \%\) of the total area is blue.) (Source: Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Banks, R... Princeton University Press.) (a) Let \(R\) denote the radius of the circumscribing circle of a five-pointed star appearing on the American flag. The star can be decomposed into 10 congruent triangles. In the figure below, \(r\) is the radius of the circumscribing circle of the pentagon in the interior of the star. Show that the area of the star is $$\begin{aligned}& \quad\quad\quad\quad\quad\quad A=\left[5 \frac{\sin A \sin B}{\sin (A+B)}\right] R^{2}\\\&\text { (Hint: }\left.\sin C=\sin \left[180^{\circ}-(A+B)\right]=\sin (A+B) .\right)\end{aligned}$$ (b) Angles \(A\) and \(B\) have values \(18^{\circ}\) and \(36^{\circ},\) respectively. Express the area of a star in terms of its radius \(R\) (c) To determine whether red or white is predominant, we consider a flag of width 10 inches, length 19 inches, length of each upper stripe 11.4 inches, and radius \(R\) of the circumscribing circle of each star 0.308 inch. The 13 stripes consist of six matching pairs of red and white stripes and one additional red, upper stripe. We must compare the area of a red, upper stripe with the total area of the 50 white stars. (i) Compute the area of the red, upper stripe. (ii) Compute the total area of the 50 white stars. (iii) Which color occupies the greatest area on the flag?

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{2 \sqrt{6}-2 i \sqrt{2}}{\sqrt{2}-i \sqrt{6}}$$

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$(3 u) \cdot v$$

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=\sin 2 \theta, r=\sqrt{2} \cos \theta ; 0 \leq \theta<\pi$$

Explain the condition that must exist to determine that there is no triangle satisfying the given values of \(a, b,\) and \(B\) once the value of \(\sin A\) is found.

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{-3 \sqrt{2}+3 i \sqrt{6}}{\sqrt{6}+i \sqrt{2}}$$

Find the area of each triangle. \(a=154\) centimeters, \(b=179\) centimeters. \(c=183\) centimeters

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$\mathbf{u} \cdot(\mathbf{v}-\mathbf{w})$$

Solve each problem.The polar equation $$r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}$$.Where \(a\) is the average distance in astronomical units from our sun and \(e\) is a constant called the eccentricity, can be used to graph the orbits of satellites of the sun. The sun will be located at the pole. The table lists \(a\) and \(e\) for the satellites.\begin{array}{|l|c|c|} \hline \text { Satellite } & a & e \\\\\hline \text { Mercury } & 0.39 & 0.206 \\\\\text { Venus } & 0.78 & 0.007 \\\\\text { Earth } & 1.00 & 0.017 \\\\\text { Mars } & 1.52 & 0.093 \\\\\text { Jupiter } & 5.20 & 0.048 \\ \text { Saturn } & 9.54 & 0.056 \\\\\text { Uranus } & 19.20 & 0.047 \\\\\text { Neptune } & 30.10 & 0.009 \\ \text { Pluto } & 39.40 & 0.249\end{array}. (A) Graph the orbits of the four closest satellites on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits. (B) Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth's distance from the sun compare with the distance from the sun to these satellites? (C) Use graphing to determine whether Pluto is always the farthest of these from the sun.

Apply the law of sines to the following: \(a=\sqrt{5}\) \(c=2 \sqrt{5}, A=30^{\circ} .\) What is the value of \(\sin C ?\) What is the measure of \(C\) ? Based on its angle measures, what kind of triangle is triangle \(A B C ?\)

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$\mathbf{u} \cdot \mathbf{v}-\mathbf{u} \cdot \mathbf{w}$$

Radio stations sometimes do not broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east- west signal, two radio towers are used, as illustrated in the figure. (IMAGE CAN'T COPY). Locations where the radio signal is received correspond to the interior of the lemniscate..$$r^{2}=40,000 \cos 2 \theta$$.Where the polar axis (or positive \(x\) -axis) points east. (A) Graph $$r^{2}=40.000 \cos 2 \theta$$. For \(0^{\circ} \leq \theta \leq 180^{\circ},\) with units in miles. Assuming that the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received. (B) Suppose a radio signal pattern is given by $$r^{2}=22,500 \sin 2 \theta$$ for \(0^{\circ} \leq \theta \leq 180^{\circ} .\) Graph this pattem and interpret the results.

Find the area of each triangle. \(a=22\) inches, \(b=45\) inches, \(c=31\) inches

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$\mathbf{u} \cdot(3 \mathbf{v})$$

Without using the law of sines, explain why no triangle \(A B C\) exists satisfying \(A=103^{\circ} 20^{\prime}, a=14.6\) feet, and \(b=20.4\) feet.

Alternating Current The alternating current in amps in an electric inductor is \(I=\frac{E}{Z}\) where \(E\) is the voltage and \(Z=R+X_{L} i\) is the impedance. If \(E=8\left(\cos 20^{\circ}+i \sin 20^{\circ}\right), R=6,\) and \(X_{L}=3\) find the current. Give the answer in rectangular form.

Find the area of each triangle. \(a=25.4\) yards, \(b=38.2\) yards, \(c=19.8\) yards

Determine whether each pair of vectors is orthogonal. $$\langle 1,2\rangle,\langle- 6,3\rangle$$

Is the graph of the function \(y=\sin x\) increasing or decreasing over the interval \(\left(0, \frac{\pi}{2}\right) ?\)

Solve each problem. The current in a circuit with voltage \(E\), resistance \(R\), capacitive reactance \(X_{c^{\prime}}\) and inductive resistance \(X_{L}\) is $$I=\frac{E}{R+\left(X_{L}-X_{c}\right) i}$$ Find \(I\) if \(E=12\left(\cos 25^{\circ}+i \sin 25^{\circ}\right), R=3, X_{L}=4\) and \(X_{c}=6 .\) Give the answer in rectangular form.

Find the area of each triangle. \(a=15.89\) inches, \(b=21.74\) inches, \(c=10.92\) inches

Determine whether each pair of vectors is orthogonal. $$\langle 3,4\rangle,\langle 6,8\rangle$$

Suppose angle \(A\) is the largest angle of an acute triangle, and let \(B\) be an angle smaller than \(A\). Explain why \(\frac{\sin B}{\sin A}<1\)

Determine whether each pair of vectors is orthogonal. $$\langle 1,0\rangle,\langle\sqrt{2}, 0\rangle$$

Solve each problem. Without actually performing the operations, state why the products $$\left[2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]\left[5\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$$ and $$\begin{array}{r}\left[2\left(\cos \left(-315^{\circ}\right)+i \sin \left(-315^{\circ}\right)\right)\right] \\ \left[5\left(\cos \left(-270^{\circ}\right)+i \sin \left(-270^{\circ}\right)\right)\right]\end{array}$$ are the same.

Solve each problem. Area of a Triangular Lot A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 meters and 21.3 meters, and the angle between them is \(42.2^{\circ} .\) What is the area of the lot?

Determine whether each pair of vectors is orthogonal. $$\langle 1,1\rangle,\langle 1,-1\rangle$$

Consider the equation \((r \text { cis } \theta)^{2}=(r \text { cis } \theta)(r \text { cis } \theta)=r^{2} \operatorname{cis}(\theta+\theta)=r^{2}\) cis \(2 \theta\) State in your own words how we can square a complex number in trigonometric form. (In the next section, we will develop this idea further.)

Solve each problem. Area of a Metal Plate A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure 16.1 meters and 15.2 meters. She knows that the angle between these sides is \(125^{\circ}\) What is the area of the surface she plans to cover with the coating?

Determine whether each pair of vectors is orthogonal. $$\sqrt{5} i-2 j,-5 i+2 \sqrt{5} j$$

Explain why no triangle \(A B C\) exists having \(A=103^{\circ}, a=12, b=13\)

Use your calculator in radian mode to find the trigonometric form of \(3+5 i .\) Approximate values to the nearest hundredth.

Solve each problem. Triangle Find the area of the Bermuda Triangle if the sides of the triangle have approximate lengths of 850 miles, 925 miles, and 1300 miles.

Determine whether each pair of vectors is orthogonal. $$-4 \mathbf{i}+3 \mathbf{j}, 8 \mathbf{i}-6 \mathbf{j}$$

Give the least positive radian measure of \(\theta\) if \(r>0\) and (a) \(r\) cis \(\theta\) has real part equal to 0 (b) \(r\) cis \(\theta\) has imaginary part equal to \(0 .\)

Solve each problem. Perfect Triangles A perfect triangle is a triangle whose sides have whole- number lengths and whose area is numerically equal to its perimeter. Show that the triangle with sides of lengths \(9,10,\) and 17 is perfect.

Under what conditions is the difference between two nonreal complex numbers \(a+b i\) and \(c+d i\) a real number?

Solve each problem. Heron Triangles A Heron triangle is a triangle having integer sides and integer area. Show that each of the following is a Heron triangle. A. \(a=11, b=13, c=20\) B. \(a=13, b=14, c=15\) C. \(a=7, b=15, c=20\)

A force of 25 pounds is required to hold an 80 -pound crate from rolling on a hill. What angle does the hill make with the horizontal?

In any triangle, the longest side is opposite the largest angle. This result from geometry was proven for acule triangles in the previous exercise set. To prove it for obtuse triangles, work Exercises \(89-92\) in order. Suppose angle \(A\) is the largest angle of an obtuse triangle. Why is \(\cos A\) negative?

A force of 18 pounds is required to hold a 60 -pound stump grinder on an incline. What angle does the incline make with the horizontal?

A force of 30 pounds is required to hold an 80 -pound pressure washer on an incline. What angle does the incline make with the horizontal?

A crate is supported by two ropes. One rope makes an angle of \(46^{\circ} 20^{\prime}\) with the horizontal and has a tension of 89.6 pounds on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.

A ship leaves port on a bearing of \(34.0^{\circ}\) and travels 10.4 miles. The ship then turns due east and travels 4.6 miles. How far is the ship from port, and what is its bearing from port?

Starting at point \(A,\) a ship sails 18.5 kilometers on a bearing of \(189^{\circ}\). then turns and sails 47.8 kilometers on a bearing of \(317^{\circ} .\) Find the distance of the ship from point \(A\).

Starting at point \(X,\) a ship sails 15.5 kilometers on a bearing of \(200^{\circ}\), then tums and sails 2.4 kilometers on a bearing of \(320^{\circ} .\) Find the distance of the ship from point \(X .\)

A motorboat sets out in the direction \(\mathrm{N} 80^{\circ} \mathrm{E}\). The speed of the boat in still water is \(20.0 \mathrm{mph}\). If the current is flowing directly south and the actual direction of the motorboat is due east, find the speed of the current and the actual speed of the motorboat.

An airline route from San Francisco to Honolulu is on a bearing of \(233.0^{\circ} .\) A jet flying at 450 mph with that heading runs into a wind blowing at 39.0 mph from a direction of \(114.0^{\circ} .\) Find the final bearing and ground speed of the plane.