Solved: Think About It What is the coordinate of any point in the plane

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((8,\ \pi/2)\) Text Transcription: (8, pi/2)

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((-2,\ 5\pi/3)\) Text Transcription: (-2, 5pi/3)

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((-4,\ -3\pi/4)\) Text Transcription: (-4, -3pi/4)

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((0,\ -7\pi/6)\) Text Transcription: (0, -7pi/6)

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((\sqrt{2},\ 2.36)\) Text Transcription: (sqrt2 , 2.36)

In Exercises 1–6, plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. \((-3,\ -1.57)\) Text Transcription: (-3, -1.57)

In Exercises 7–10, use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. \((7,\ 5\pi/4)\) Text Transcription: (7, 5pi/4)

In Exercises 7–10, use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. \((-2,\ 11\pi/6)\) Text Transcription: (-2, 11pi/6)

In Exercises 7–10, use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. (-4.5, 3.5)

In Exercises 7–10, use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. (9.25, 1.2)

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). (2, 2) Text Transcription: 0 leq theta < 2pi

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). (0, -6) Text Transcription: 0 leq theta < 2pi

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). (3, -4) Text Transcription: 0 leq theta < 2pi

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). (4, -2) Text Transcription: 0 leq theta < 2pi

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). \((-1,\ -\sqrt{3})\) Text Transcription: 0 leq theta < 2pi (-1, -sqrt3)

In Exercises 11–16, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0\ \leq\ \theta\ <\ 2\pi\). \((3,\ -\sqrt{3})\) Text Transcription: 0 leq theta < 2pi (3, -sqrt3)

In Exercises 17–20, use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (3, -2)

In Exercises 17–20, use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \((3\sqrt{2},\ 3\sqrt{2})\) Text Transcription: (3sqrt2, 3sqrt2)

In Exercises 17–20, use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \((\frac{7}{4},\ \frac{5}{2})\) Text Transcription: (7/4 , 5/2)

In Exercises 17–20, use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (0, -5)

Plot the point (4, 3.5) if the point is given in (a) rectangular coordinates and (b) polar coordinates.

Graphical Reasoning (a) Set the window format of a graphing utility to rectangular coordinates and locate the cursor at any position off the axes. Move the cursor horizontally and vertically. Describe any changes in the displayed coordinates of the points. (b) Set the window format of a graphing utility to polar coordinates and locate the cursor at any position off the axes. Move the cursor horizontally and vertically. Describe any changes in the displayed coordinates of the points. (c) Why are the results in parts (a) and (b) different?

In Exercises 23–26, match the graph with its polar equation. [The graphs are labeled (a), (b), (c), and (d).] \(r=2 \sin \theta\) Text Transcription: r=2 sin theta

In Exercises 23–26, match the graph with its polar equation. [The graphs are labeled (a), (b), (c), and (d).] \(r=4 \cos 2 \theta\) Text Transcription: r=4 cos 2 theta

In Exercises 23–26, match the graph with its polar equation. [The graphs are labeled (a), (b), (c), and (d).] \(r=3(1+ \cos \theta)\) Text Transcription: r=3(1+cos theta)

In Exercises 23–26, match the graph with its polar equation. [The graphs are labeled (a), (b), (c), and (d).] \(r=2 \sec \theta\) Text Transcription: r=2 sec theta

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \(x^2+y^2=9\) Text Transcription: x^2 +y^2 =9

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \(x^2-y^2=9\) Text Transcription: x^2 -y^2 =9

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \(x^2+y^2=a^2\) Text Transcription: x^2 +y^2 =a^2

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \(x^2 +y^2-2ax=0\) Text Transcription: x^2 +y^2 -2ax=0

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. y = 8

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. x = 12

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. 3x - y + 2 = 0

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. xy = 4

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \(y^2=9x\) Text Transcription: y^2 =9x

In Exercises 27–36, convert the rectangular equation to polar form and sketch its graph. \((x^2+y^2)^2-9(x^2-y^2)=0\) Text Transcription: (x^2 +y^2)^2 -9(x^2 -y^2)=0

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. r = 4

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. r = -5

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=3 \sin \theta\) Text Transcription: r = 3 sin theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=5 \cos \theta\) Text Transcription: r=5 cos theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=\theta\) Text Transcription: r=theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(\theta=\frac{5\pi}{6}\) Text Transcription: theta=5pi/6

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=3 \sec \theta\) Text Transcription: r=3 sec theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=2 \csc \theta\) Text Transcription: r = 2 csc theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=\sec \theta \tan \theta\) Text Transcription: r=sec theta tan theta

In Exercises 37– 46, convert the polar equation to rectangular form and sketch its graph. \(r=\cot \theta \csc \theta\) Text Transcription: r=cot theta csc theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=2-5 \cos \theta\) Text Transcription: theta r=2-5 cos theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=3(1-4 \cos \theta)\) Text Transcription: theta r=3(1-4 cos theta)

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=2+\sin \theta\) Text Transcription: theta r=2+sin theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=4+3 \cos \theta\) Text Transcription: theta r=4+3 cos theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=\frac{2}{1+\cos \theta}\) Text Transcription: theta r=2/1+cos theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=\frac{2}{4-3 \sin \theta}\) Text Transcription: theta r=2/4-3 sin theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=2 \cos (\frac{3\theta}{2})\) Text Transcription: theta r=2 cos (3theta/2)

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r=3 \sin (\frac{5\theta}{2})\) Text Transcription: theta r=3 sin (5 theta/2)

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r^2=4 \sin 2\theta\) Text Transcription: theta r^2 = 4 sin 2 theta

In Exercises 47–56, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. \(r^2=\frac{1}{\theta}\) Text Transcription: theta r^2 =1/theta

Convert the equation \(r=2(h \cos \theta + k \sin \theta)\) to rectangular form and verify that it is the equation of a circle. Find the radius and the rectangular coordinates of the center of the circle. Text Transcription: r=2(h cos theta + k sin theta)

Distance Formula (a) Verify that the Distance Formula for the distance between the two points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) in polar coordinates is \(d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\). (b) Describe the positions of the points relative to each other for \(\theta_{1}=\theta_{2}\). Simplify the Distance Formula for this case. Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\). Is the simplification what you expected? Explain. (d) Choose two points on the polar coordinate system and find the distance between them. Then choose different polar representations of the same two points and apply the Distance Formula again. Discuss the result. Text Transcription: (r_1,theta_1) (r_2,theta_2) d=sqrt r_1^2+r_2^2-2r_1r_2 cos(theta_1-theta_2) theta_1=theta_2 theta_1-theta_2=90^o

In Exercises 59-62, use the result of Exercise 58 to approximate the distance between the two points in polar coordinates. \(\left(1, \frac{5 \pi}{6}\right),\left(4, \frac{\pi}{3}\right)\) Text Transcription: (1,5 pi/6), (4, pi/3)

In Exercises 59-62, use the result of Exercise 58 to approximate the distance between the two points in polar coordinates. \(\left(8, \frac{7 \pi}{4}\right)\), \((5, \pi)\) Text Transcription: (8,7pi/4) (5,pi)

In Exercises 59-62, use the result of Exercise 58 to approximate the distance between the two points in polar coordinates. (2, 0.5), (7, 1.2)

In Exercises 59-62, use the result of Exercise 58 to approximate the distance between the two points in polar coordinates. (4, 2.5), (12, 1)

In Exercises 63 and 64 , find dy/dx and the slopes of the tangent lines shown on the graph of the polar equation. \(r=2+3 \sin \theta\) Text Transcription: r=2+3 sin theta

In Exercises 63 and 64 , find dy/dx and the slopes of the tangent lines shown on the graph of the polar equation. \(r=2(1-\sin \theta)\) Text Transcription: r=2(1-sin theta)

In Exercises 65-68, use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of \(\theta\), and (c) find dy/dx at the given value of \(\theta\). (Hint: Let the increment between the values of \(\theta\) equal \(\pi / 24\).) \(r=3(1-\cos \theta)\), \(\theta=\frac{\pi}{2}\) Text Transcription: theta pi/24 r=3(1-cos theta) theta=pi/2

In Exercises 65-68, use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of \(\theta\), and (c) find dy/dx at the given value of \(\theta\). (Hint: Let the increment between the values of \(\theta\) equal \(\pi / 24\).) \(r=3-2 \cos \theta, \theta=0\) Text Transcription: theta pi/24 r=3-2 cos theta, theta=0

In Exercises 65-68, use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of \(\theta\), and (c) find dy/dx at the given value of \(\theta\). (Hint: Let the increment between the values of \(\theta\) equal \(\pi / 24\).) \(r=3 \sin \theta\), \(\theta=\frac{\pi}{3}\) Text Transcription: theta pi/24 r=3 sin theta theta=pi/3

In Exercises 65-68, use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of \(\theta\), and (c) find dy/dx at the given value of \(\theta\). (Hint: Let the increment between the values of \(\theta\) equal \(\pi / 24\).) r = 4, \(\theta=\frac{\pi}{4}\) Text Transcription: theta pi/24 theta=pi/4

In Exercises 69 and 70 , find the points of horizontal and vertical tangency (if any) to the polar curve. \(r=1-\sin \theta\) Text Transcription: r=1-sin theta

In Exercises 69 and 70 , find the points of horizontal and vertical tangency (if any) to the polar curve. \(r=a \sin \theta\) Text Transcription: r=a sin theta

In Exercises 71 and 72 , find the points of horizontal tangency (if any) to the polar curve. \(r=2 \csc \theta+3\) Text Transcription: r=2 csc theta+3

In Exercises 71 and 72 , find the points of horizontal tangency (if any) to the polar curve. \(r=a \sin \theta \cos ^{2} \theta\) Text Transcription: r=a sin theta cos^2 theta

In Exercises 73-76, use a graphing utility to graph the polar equation and find all points of horizontal tangency. \(r=4 \sin \theta \cos ^{2} \theta\) Text Transcription: r=4 sin theta cos^2 theta

In Exercises 73-76, use a graphing utility to graph the polar equation and find all points of horizontal tangency. \(r=3 \cos 2 \theta \sec \theta\) Text Transcription: r=3 cos 2 theta sec theta

In Exercises 73-76, use a graphing utility to graph the polar equation and find all points of horizontal tangency. \(r=2 \csc \theta+5\) Text Transcription: r=2 csc theta+5

In Exercises 73-76, use a graphing utility to graph the polar equation and find all points of horizontal tangency. \(r=2 \cos (3 \theta-2)\) Text Transcription: r=2 cos (3 theta-2)

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=5 \sin \theta\) Text Transcription: r=5 sin theta

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=5 \cos \theta\) Text Transcription: r=5 cos theta

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=2(1-\sin \theta)\) Text Transcription: r=2(1-sin theta)

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=3(1-\cos \theta)\) Text Transcription: r=3(1-cos theta)

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=4 \cos 3 \theta\) Text Transcription: r=4 cos 3 theta

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=-\sin 5 \theta\) Text Transcription: r=-sin 5 theta

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=3 \sin 2 \theta\) Text Transcription: r=3 sin 2 theta

In Exercises 77-84, sketch a graph of the polar equation and find the tangents at the pole. \(r=3 \cos 2 \theta\) Text Transcription: r=3 cos 2 theta

In Exercises 85-96, sketch a graph of the polar equation. r = 8

In Exercises 85-96, sketch a graph of the polar equation. r = 1

In Exercises 85-96, sketch a graph of the polar equation. \(r=4(1+\cos \theta)\) Text Transcription: r=4(1+cos theta)

In Exercises 85-96, sketch a graph of the polar equation. \(r=1+\sin \theta\) Text Transcription: r=1+sin theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=3-2 \cos \theta\) Text Transcription: r=3-2 cos theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=5-4 \sin \theta\) Text Transcription: r=5-4 sin theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=3 \csc \theta\) Text Transcription: r=3 csc theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) Text Transcription: r=6/2 \sin theta-3 cos theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=2 \theta\) Text Transcription: r=2 theta

In Exercises 85-96, sketch a graph of the polar equation. \(r=\frac{1}{\theta}\) Text Transcription: r=1/theta

In Exercises 85-96, sketch a graph of the polar equation. \(r^{2}=4 \cos 2 \theta\) Text Transcription: r^2=4 cos 2 theta

In Exercises 85-96, sketch a graph of the polar equation. \(r^{2}=4 \sin \theta\) Text Transcription: r^2=4 sin theta

In Exercises 97-100, use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Polar Equation Asymptote Conchoid \(r=2-\sec \theta\) x = -1 Text Transcription: r=2-sec theta

In Exercises 97-100, use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Polar Equation Asymptote Conchoid \(r=2+\csc \theta\) y = 1 Text Transcription: r=2+csc theta

In Exercises 97-100, use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Polar Equation Asymptote Hyperbolic spiral \(r=2/\theta\) y = 2 Text Transcription: r=2/theta

In Exercises 97-100, use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Polar Equation Asymptote Strophoid \(r=2 \cos 2 \theta \sec \theta\) x = -2 Text Transcription: r=2 cos 2 theta sec theta

Describe the differences between the rectangular coordinate system and the polar coordinate system.

Give the equations for the coordinate conversion from rectangular to polar coordinates and vice versa.

How are the slopes of tangent lines determined in polar coordinates? What are tangent lines at the pole and how are they determined?

Describe the graphs of the following polar equations. (a) r = 7 (b) \(r^{2}=7\) (c) \(r=\frac{7}{\cos \theta}\) (d) \(r=\frac{7}{\sin \theta}\) (e) \(r=7 \cos \theta\) (f) \(r=7 \sin \theta\) Text Transcription: r^2=7 r=7/cos theta r=7/sin theta r=7 cos theta r=7 sin theta

Sketch the graph of \(r=4 \sin \theta\) over each interval. (a) \(0 \leq \theta \leq \frac{\pi}{2}\) (b) \(\frac{\pi}{2} \leq \theta \leq \pi\) (c) \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) Text Transcription: r=4 sin theta 0<=theta<=pi/2 pi/2<=theta<=pi -pi/2<=theta<=pi/2

Think About It Use a graphing utility to graph the polar equation \(r=6[1+\cos (\theta-\phi)]\) for (a) \(\phi=0\), (b) \(\phi=\pi / 4\), and (c) \(\phi=\pi / 2\). Use the graphs to describe the effect of the angle \(\phi\). Write the equation as a function of \(\sin \theta\) for part (c). Text Transcription: r=[6+cos(theta-phi)] phi=0 phi=pi/4 phi=pi/2 phi sin theta

Verify that if the curve whose polar equation is \(r=f(\theta)\) is rotated about the pole through an angle \(\phi\), then an equation for the rotated curve is \(r=f(\theta-\phi)\). Text Transcription: r=f(theta) Phi r=f(theta-phi)

The polar form of an equation of a curve is \(r=f(\sin \theta)\). Show that the form becomes (a) \(r=f(-\cos \theta)\) if the curve is rotated counterclockwise \(\pi / 2\) radians about the pole. (b) \(r=f(-\sin \theta)\) if the curve is rotated counterclockwise \(\pi\) radians about the pole. (c) \(r=f(\cos \theta)\) if the curve is rotated counterclockwise \(3 \pi / 2\) radians about the pole. Text Transcription: r=f(sin theta) r=f(-cos theta) pi/2 r=f(-sin theta) pi r=f(cos theta) 3pi/2

In Exercises 109-112, use the results of Exercises 107 and 108. Write an equation for the limaçon \(r=2-\sin \theta\) after it has been rotated by the given amount. Use a graphing utility to graph the rotated limaçon. (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\pi\) (d) \(\frac{3 \pi}{2}\) Text Transcription: r=2-sin theta pi/4 pi/2 pi 3 pi/2

In Exercises 109-112, use the results of Exercises 107 and 108. Write an equation for the rose curve \(r=2 \sin 2 \theta\) after it has been rotated by the given amount. Verify the results by using a graphing utility to graph the rotated rose curve. (a) \(\frac{\pi}{6}\) (b) \(\frac{\pi}{2}\) (c) \(\frac{2 \pi}{3}\) (d) \(\pi\) Text Transcription: r=2 sin 2 theta pi/6 pi/2 2 pi/3 pi

In Exercises 109-112, use the results of Exercises 107 and 108. Sketch the graph of each equation. (a) \(r=1-\sin \theta\) (b) \(r=1-\sin \left(\theta - \frac{\pi}{4}\right)\) Text Transcription: r=1-sin theta r=1-sin(theta-pi/4)

In Exercises 109-112, use the results of Exercises 107 and 108. Prove that the tangent of the angle \(\psi(0 \leq \psi \leq \pi / 2)\) between the radial line and the tangent line at the point \((r, \theta)\) on the graph of \(r=f(\theta)\) (see figure) is given by \(\tan \psi=|r /(d r / d \theta)|\). Text Transcription: psi(0<=psi<=pi/2) (r,theta) r=f(theta) Tan psi=|r/(dr/dtheta)|

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) \(r=2(1-\cos \theta)\) \(\theta=\pi\) Text Transcription: psi theta r=2(1-cos theta) theta=pi

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) \(r=3(1-\cos \theta)\) \(\theta=3 \pi / 4\) Text Transcription: psi theta r=3(1-cos theta) theta=3 pi/4

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) \(r=2 \cos 3 \theta\) \(\theta=\pi / 4\) Text Transcription: psi theta r=2 cos 3 theta theta=pi/4

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) \(r=4 \sin 2 \theta\) \(\theta=\pi / 6\) Text Transcription: psi theta r=4 sin theta theta=pi/6

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) \(r=\frac{6}{1-\cos \theta}\) \(\theta=2 \pi / 3\) Text Transcription: psi theta r=6/1-cos theta theta=2 pi/3

In Exercises 113-118, use the result of Exercise 112 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta\). Identify the angle \(\psi\). Polar Equation Value of \(\theta\) r = 5 \(\theta=\pi / 6\) Text Transcription: psi theta theta=pi/6

True or False? In Exercises 119-122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point on the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\). Text Transcription: (r_1,theta_1) (r_2,theta_2) |r_1|=|r_2|

True or False? In Exercises 119-122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point on the polar coordinate system, then \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer n. Text Transcription: (r,theta_1) (r,theta_2) theta_1=theta_2+2 pi n

True or False? In Exercises 119-122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If x > 0, then the point (x, y) on the rectangular coordinate system can be represented by \((r, \theta)\) on the polar coordinate system, where \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta=\arctan (y / x)\). Text Transcription: (r,theta) r=sqrt x^2+y^2 theta=arctan(y/x)

True or False? In Exercises 119-122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The polar equations \(r=\sin 2 \theta\), \(r=-\sin 2 \theta\), and \(r=\sin (-2 \theta)\) all have the same graph. Text Transcription: r=sin 2 theta r=-sin 2 theta r=sin(-2 theta)