Chapter 9

Graph each point in a polar coordinate system then convert the given polar coordinates to rectangular coordinates. (a) \(\left(3, \frac{2 \pi}{3}\right)\) (b) \(\left(4, \frac{11 \pi}{6}\right)\) (c) \(\left(4,-\frac{\pi}{6}\right)\)

In the same picture, graph the four polar equations \(r=2\) \(r=4, r=6,\) and \(r=8 .\) Describe the graphs.

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=2-4 t, y=3-5 t ; t=0$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle 4,3\rangle$$

Graph each point in a polar coordinate system then convert the given polar coordinates to rectangular coordinates. (a) \(\left(5, \frac{\pi}{4}\right)\) (b) \(\left(-5, \frac{\pi}{4}\right)\) (c) \(\left(-5,-\frac{\pi}{4}\right)\)

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=3-t^{2}, y=4+t^{3} ; t=-1$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle 5,12\rangle$$

Graph each point in a polar coordinate system then convert the given polar coordinates to rectangular coordinates. (a) \(\left(1, \frac{\pi}{2}\right)\) (b) \(\left(1, \frac{5 \pi}{2}\right)\) (c) \(\left(-1, \frac{\pi}{8}\right)\)

Graph the polar equations. $$r=\theta /(2 \pi), \text { for } \theta \geq 0$$

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=5 \cos t, y=2 \sin t ; t=\pi / 6$$

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. $$\overrightarrow{S Q}$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle-4,2\rangle$$

onvert the given rectangular coordinates to polar coordinates. Express your answers in such a way that ris nonnegative and \(0 \leq \theta<2 \pi\) $$(3, \sqrt{3})$$

Graph the polar equations. $$r=\theta / \pi, \text { for }-4 \pi \leq \theta \leq 0$$

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=4 \cos 2 t, y=6 \sin 2 t ; t=\pi / 3$$

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. $$\overrightarrow{Q S}$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle-6,-6\rangle$$

onvert the given rectangular coordinates to polar coordinates. Express your answers in such a way that ris nonnegative and \(0 \leq \theta<2 \pi\) $$(-1,-1)$$

Graph the polar equations. $$r=\ln \theta, \text { for } 1 \leq \theta \leq 3 \pi$$

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=3 \sin ^{3} t, y=3 \cos ^{3} t ; t=\pi / 4$$

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. $$\overrightarrow{O P}$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\left\langle\frac{3}{4},-\frac{1}{2}\right\rangle$$

onvert the given rectangular coordinates to polar coordinates. Express your answers in such a way that ris nonnegative and \(0 \leq \theta<2 \pi\) $$(0,-2)$$

Graph the polar equations. (a) \(r=e^{\theta / 2 \pi},\) for \(0 \leq \theta \leq 2 \pi\) (b) \(r=e^{-\theta / 2 \pi},\) for \(0 \leq \theta \leq 2 \pi\)

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=\sin t-\sin 2 t, y=\cos t+\cos 2 t ; t=2 \pi / 3$$

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. $$\overrightarrow{P O}$$

Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle-3,0\rangle$$

Convert to rectangular form. $$r=2 \cos \theta$$

A ladder 18 ft long leans against a building. The ladder forms an angle of \(60^{\circ}\) with the ground. (a) How high up the side of the building does the ladder reach? [Here and in part (b), give two forms for your answers: one with radicals and one (using a calculator) with decimals, rounded to two places.] (b) Find the horizontal distance from the foot of the ladder to the base of the building.

Graph the polar equations. \(r=\theta(\text {spiral of Archimedes})\) Suggestion: Use a viewing rectangle extending from -30 to 30 in both the \(x\) - and \(y\) -directions. Let \(\theta\) run from 0 to \(2 \pi,\) then from 0 to \(4 \pi\) and finally from 0 to \(8 \pi\)

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. $$\overrightarrow{P Q}+\overrightarrow{Q S}$$

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(2,3) \text { and } Q(3,7)$$

From a point level with and 1000 ft away from the base of the Washington Monument, the angle of elevation to the top of the monument is \(29.05^{\circ} .\) Determine the height of the monument to the nearest half foot.

Graph the polar equations. \(r=1 / \theta\) (hyperbolic spiral) Suggestion: Use a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. Let \(\theta\) run from 0 to \(2 \pi,\) then from 0 to \(4 \pi\) and finally from 0 to \(8 \pi\)

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=t+1, y=t^{2}$$

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. $$\overrightarrow{S Q}+\overrightarrow{Q P}$$

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(5,1) \text { and } Q(4,9)$$

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=2 t-1, y=t^{2}-1$$

Use degree measure for your answers. In parts (c) and (d), use a calculator and round the results to one decimal place. (a) In \(\triangle A B C, \sin B=\sqrt{2} / 2 .\) What are the possible values for \(\angle B ?\) (b) In \(\triangle D E F, \cos E=\sqrt{2} / 2 .\) What are the possible values for \(\angle E ?\) (c) In \(\triangle G H I, \sin H=1 / 4 .\) What are the possible values for \(\angle H ?\) (d) In \(\triangle J K L, \cos K=-2 / 3 .\) What are the possible values for \(\angle K ?\)

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. $$\overrightarrow{O P}+\overrightarrow{P Q}$$

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(-2,-3) \text { and } Q(-3,-2)$$

In isosceles triangle \(A B C\), the sides are of length \(A C=B C=8\) and \(A B=4 .\) Find the angles of the triangle. Express the answers both in radians, rounded to two decimal places, and in degrees, rounded to one decimal place. Hints: To find \(\angle A,\) start by drawing an altitude from \(C\) to side \(\overline{A B}\). Then for \(\angle C\), use the fact that the sum of the angles in a triangle is \(\pi\) radians or \(180^{\circ}\).

Graph the polar equations. \(r=\sqrt{\theta}(\text {spiral of Fermat})\) Suggestion: Use a viewing rectangle extending from -5 to 5 in both the \(x\) - and \(y\) -directions. Let \(\theta\) run from \(\theta\) to \(2 \pi,\) then from 0 to \(4 \pi\) and finally from 0 to 8 \(\pi\)

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=t^{2}-1, y=t+1$$

Use degree measure for your answers. In parts (c) and (d), use a calculator and round the results to one decimal place. (a) In \(\triangle A B C, \sin B=\sqrt{3} / 2 .\) What are the possible values for \(\angle B ?\) (b) In \(\triangle D E F, \cos E=-\sqrt{3} / 2 .\) What are the possible values for \(\angle E ?\) (c) In \(\triangle G H I, \sin H=2 / 9 .\) What are the possible values for \(\angle H ?\) (d) In \(\triangle J K L, \cos K=2 / 3 .\) What are the possible values

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. $$\overrightarrow{O S}+\overrightarrow{S Q}$$

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(0,-4) \text { and } Q(0,-8)$$

Convert to rectangular form. $$r=3 \cos 2 \theta$$

Graph the polar equations. $$r=1+\cos \theta$$

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=t-4, y=|t|$$