Chapter 9

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{O P}+\overrightarrow{O Q}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. (a) \(x=3 t+2, y=3 t-2,(-2 \leq t \leq 2)\) (b) \(x=3 t+2, y=3 t-2,(-3 \leq t \leq 3)\)

Convert to polar form. $$9 x^{2}+y^{2}=9$$

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$3 a+4 a$$

Graph the polar equations. $$r=\cos 2 \theta(\text {four-leafed rose})$$

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{R P}+\overrightarrow{R S}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\ln (3 t+2), y=\ln (3 t-2)\) using: (a) \(2 / 3

Convert to polar form. $$x^{2}\left(x^{2}+y^{2}\right)=y^{2}$$

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$|4 \mathbf{b}+5 \mathbf{b}|$$

Graph the polar equations. $$r=2 \sin 2 \theta(\text {four-leafed rose})$$

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{Q P}+\overrightarrow{Q R}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. $$x=4 \cos t, y=3 \sin t, 0 \leq t \leq 2 \pi(\text {ellipse})$$

Use the given information to find the cosine of each angle in \(\triangle \overline{A B C}\) \(a=6 \mathrm{cm}, b=7 \mathrm{cm}, c=10 \mathrm{cm}\)

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\mathbf{a}-\mathbf{b}$$

Graph the polar equations. $$r=\sin 3 \theta(\text {three-leafed rose})$$

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{S O}+\overrightarrow{S Q}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=4 \cos t, y=-3 \sin t, 0 \leq t \leq 2 \pi(\) the same as the ellipse in Exercise 25 but traced out in the opposite direction)

Use the given information to find the cosine of each angle in \(\triangle \overline{A B C}\) \(a=17 \mathrm{cm}, b=8 \mathrm{cm}, c=15 \mathrm{cm}\) (For this particular tri- angle, you can check your answers, because there is an alternative method of solution that does not require the law of cosines.)

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\mathbf{b}-\mathbf{c}$$

Graph the polar equations. \(r=2 \cos 5 \theta\) (five-leafed rose)

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{S Q}+\overrightarrow{S R}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=4 \cos t, y=3 \sin t, 0 \leq t \leq \pi / 2\) (one-quarter of an ellipse)

Compute each angle of the given triangle. Where necessary, use a calculator and round to one decimal place. $$a=7, b=8, c=13$$

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$3 b-4 d$$

Graph the polar equations. $$r=4+2 \sin \theta(\text {limacon with no inner loop})$$

The vectors \(\mathbf{F}\) and \(\mathbf{G}\) denote two forces that act on an object: G acts horizontally to the right, and \(\mathbf{F}\)acts vertically upward. In each case, use the information that is given to compute \(|\mathbf{F}+\mathbf{G}|\) and \(\theta,\) where \(\theta\) is the angle between \(\mathbf{G}\) and the resultant. $$|\mathbf{F}|=4 \mathbf{N},|\mathbf{G}|=5 \mathbf{N}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. Figure cannot copy (a) The curve in the accompanying figure is called an astroid (or a hypocycloid of four cusps). A pair of parametric equations for the curve is \(x=\cos ^{3} t\) \(y=\sin ^{3} t .\) By eliminating the parameter \(t,\) find the \(x-y\) equation for the curve. Hint: In each equation, raise both sides to the two-thirds power. (b) Graph the curve in part (a) for \(0 \leq t \leq 2 \pi\) to reproduce the figure above. [The hypocycloid of four cusps was first studied by the Danish astronomer Olaf Roemer \((1644-1710)\) and by the Swiss mathematician Jacob Bernoulli ( \(1654-1705 \text { ). }]\)

In triangle \(O A B,\) lengths \(O A=O B=6\) in. and \(\angle A O B=72^{\circ} .\) Find \(A B .\) Ilint: Draw a perpendicular from \(O\) to \(A B\). Round the answer to one decimal place.

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\frac{1}{|3 \mathbf{b}-4 \mathbf{d}|}(3 \mathbf{b}-4 \mathbf{a})$$

Graph the polar equations. $$r=1.5-\cos \theta(\text {limaçon with no inner loop })$$

The vectors \(\mathbf{F}\) and \(\mathbf{G}\) denote two forces that act on an object: G acts horizontally to the right, and \(\mathbf{F}\)acts vertically upward. In each case, use the information that is given to compute \(|\mathbf{F}+\mathbf{G}|\) and \(\theta,\) where \(\theta\) is the angle between \(\mathbf{G}\) and the resultant. $$|\mathbf{F}|=15 \mathbf{N},|\mathbf{G}|=6 \mathbf{N}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=2 \cos t+\cos 2 t, y=2 \sin t-\sin 2 t, \quad 0 \leq t \leq 2 \pi[\) This curve is the deltoid. It was first studied by the Swiss mathematician Leonhard Euler \((1707-1783) .]\)

Compute each angle of the given triangle. Where necessary, use a calculator and round to one decimal place. $$a=b=2 / \sqrt{3}, c=2$$

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\mathbf{a}-(\mathbf{b}+\mathbf{c})$$

Graph the polar equations. \(r=8\) tan \(\theta\) (kappa curve)

The vectors \(\mathbf{F}\) and \(\mathbf{G}\) denote two forces that act on an object: G acts horizontally to the right, and \(\mathbf{F}\)acts vertically upward. In each case, use the information that is given to compute \(|\mathbf{F}+\mathbf{G}|\) and \(\theta,\) where \(\theta\) is the angle between \(\mathbf{G}\) and the resultant. $$|\mathbf{F}|=|\mathbf{G}|=9 \mathbf{N}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\frac{\cos t}{1+\sin ^{2} t}, y=\frac{\sin t \cos t}{1+\sin ^{2} t}, \quad 0 \leq t \leq 2 \pi\) [This curve is the lemniscate of Bernoulli. The Swiss mathematician Jacob Bernoulli ( \(1654-1705\) ) studied the curve and took the name lemniscate from the Greek lemniskos, meaning "ribbon." \(]\)

Compute each angle of the given triangle. Where necessary, use a calculator and round to one decimal place. $$a=36, b=77, c=85$$

An observer in a lighthouse is \(66 \mathrm{ft}\) above the surface of the water. The observer sees a ship and finds the angle of depression to be \(0.7^{\circ} .\) Estimate the distance of the ship from the base of the lighthouse. Round the answer to the nearest 5 feet.

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$(\mathbf{a}-\mathbf{b})-\mathbf{c}$$

The vectors \(\mathbf{F}\) and \(\mathbf{G}\) denote two forces that act on an object: G acts horizontally to the right, and \(\mathbf{F}\)acts vertically upward. In each case, use the information that is given to compute \(|\mathbf{F}+\mathbf{G}|\) and \(\theta,\) where \(\theta\) is the angle between \(\mathbf{G}\) and the resultant. $$|\mathbf{F}|=28 \mathbf{N},|\mathbf{G}|=1 \mathbf{N}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=2 \tan t, y=2 \cos ^{2} t, \quad 0 \leq t \leq 2 \pi \quad\) Remark: If you eliminate the parameter \(t\), you'll find that the Cartesian form of the curve is \(y=8 /\left(x^{2}+4\right) .\) (Verify this last statement, first algebraically, then graphically.) The curve is known as the witch of Agnesi, named after the Italian mathematician and scientist Maria Gaetana Agnesi ( \(1718-\) 1799)\(.\) The word "witch" in the name of the curve is the result of a mistranslation from Italian to English. In Agnesi's time, the curve was known as la versiera, an Italian name with a Latin root meaning "to turn." In translation, the word versiera was confused with another Italian word avversiera, which means "wife of the devil" or "witch."

Round each answer to one decimal place. A regular pentagon is inscribed in a circle of radius 1 unit. Find the perimeter of the pentagon. Hint: First find the length of a side using the law of cosines.

From a point on ground level, you measure the angle of clevation to the top of a mountain to be \(38^{\circ} .\) Then you walk 200 m farther away from the mountain and find that the angle of elevation is now \(20^{\circ}\). Find the height of the mountain. Round the answer to the nearest meter.

Compute the distance between the given points. (The coordinates are polar coordinates.) $$\left(2, \frac{2 \pi}{3}\right) \text { and }\left(4, \frac{\pi}{6}\right)$$

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$|\mathbf{c}+\mathbf{d}|^{2}-|\mathbf{c}-\mathbf{d}|^{2}$$

(a) Use one of the polar symmetry tests to show that the graph of \(r=\cos ^{2} \theta-2 \cos \theta\) is symmetric about the \(x\) -axis. (b) Graph the equation given in part (a) and note that the curve is indeed symmetric about the \(x\) -axis.

The vectors \(\mathbf{F}\) and \(\mathbf{G}\) denote two forces that act on an object: G acts horizontally to the right, and \(\mathbf{F}\)acts vertically upward. In each case, use the information that is given to compute \(|\mathbf{F}+\mathbf{G}|\) and \(\theta,\) where \(\theta\) is the angle between \(\mathbf{G}\) and the resultant. $$|\mathbf{F}|=3.22 \mathbf{N},|\mathbf{G}|=7.21 \mathbf{N}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}, \quad-\infty

Round each answer to one decimal place. Find the perimeter of a regular nine-sided polygon inscribed in a circle of radius \(4 \mathrm{cm}\). (See the hint for Exercise \(31 .)\)