To get these, if the first number (\(r\)) is negative, you want to go in the opposite direction, and if the angle is negative, you want to go clockwise instead of counterclockwise from the positive \(x\)–axis. You can see how different these graphs look: To convert Rectangular Equations to Polar Equations, we want to get rid of the \(x\)’s and \(y\)’s and only have \(r\)’s and/or \(\theta \)’s in the answer. plot.axes(projection='polar') # Set the title of the polar plot plot.title('Circle in polar format:r=R') # Plot a circle with radius 2 using polar form rads = np.arange(0, … The first distinction to be made is between when n is an even or odd number. There are many familiar shapes such as lines, circles, parabolas, and ellipses which can be expressed in polar form. (Note that you can also put these in your graphing calculator, as an example, with radians: MODE: RADIAN, POLAR and WINDOW:  θ = [0, 2π], θstep =  π/12 or π/6, X = [–10, 10], Y = [–6, 6], and then using “Y=” to put in the equation, or just put in graph and use ZOOM ZTRIG (option 7). Note that since we have the starting point for these graphs, and the distance between the petals, the t-chart isn’t that helpful. The reason these points are “phantom” is because, although we don’t necessarily get them algebraically, we can see them on a graph. Of course, I can only discover so many of them, but this is a page of really cool graphs that I have found. (Remember that 240° and –120°, and 60° and –240° are co-terminal angles). Note that we talk about converting back and forth from Polar Complex Form to Rectangular Complex form here in the Trigonometry and the Complex Plane section. See also Equiangular Spiral. Just as a quick review, the polar coordinate system is very similar to that of the rectangular coordinate system. No. In what order are the petals drawn? Since the graph hits 5 (\(a\)) on the \(\boldsymbol {y}\)-axis and goes out to 9 to the left, we have \(r=5-4\cos \theta \), since \(b=9-a\). For converting back to polar, make sure answers are either between 0 and 360° for degrees or 0 to \(2\pi \) for radians. Thank you in advanced. .large-mobile-banner-1-multi{display:block !important;float:none;line-height:0px;margin-bottom:15px !important;margin-left:0px !important;margin-right:0px !important;margin-top:15px !important;min-height:250px;min-width:970px;text-align:center !important;}eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_7',127,'0','0']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_8',127,'0','1']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_9',127,'0','2']));Let’s start with polar equations that result in circle graphs: Note:  This makes sense since \(r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}\), and the equation of a circle is \({{x}^{2}}+{{y}^{2}}={{5}^{2}}\). In particular, there are three cases : |a| = |b|. Use GraphFunc utility online to sketch the following polar graphs and find its derivatives at. They are in the form \(r=a+b\cos \theta \) or \(r=a+b\sin \theta \). A polar chart represents data along radial and angular axes. Note also that after we solve for one variable (like \(\theta \)), we have to plug it back in either equation to get the other coordinate (like \(r\)). (Note that since the t-chart starts on the positive \(x\)-axis, the \(r\)’s are negative in the chart). Note: For a rose graph, you may be asked to name the order that petals are drawn. It can have up to 6 equations. With cos, graph is horizontal across the \(\boldsymbol {x}\)-axis. If cos is negative (such as \(r=3-3\cos \theta \)), reflect over \(y\)-axis so it’s on left-hand side. Also, since \(a-b \,\,(5-4=1)\) is where it hits the \(x\)-axis, it looks good! \(\begin{array}{l}r=a+b\cos \theta ,\,\,\,a>b\\r=a+b\sin \theta ,\,\,\,a>b\end{array}\). By putting in smaller values of θstep, such as .1, the graph is drawn more slowly and more accurately; to redraw graph, you can turn the graph off and back on by going to “=” and un-highlighting and highlighting it back again before hitting “graph”. Plotly Python Open Source Graphing Library Scientific Charts. So, join me on a Polar Graph hunt! This is because, with an “\(r\)” of 0, the \(\theta \) could really be anything, since we aren’t going out any distance. Let’s first convert from polar to rectangular form; to do this we use the following formulas, as we can see this from the graph: This conversion is pretty straight-forward: \(\begin{array}{l}x\,\,=\,\,r\,\cos \,\theta \\y\,\,=\,\,r\,\sin \,\theta \end{array}\). Here is something that looks a little more rose-like. You can also use the graphing calculator as shown above, but make the θstep smaller to slow down the drawing of the graph. If \(\theta <0\), you go clockwise with the angle, starting with the positive \(x\)–axis. The Archimedean Spiral The Archimedean spiral is formed from the equation r = aθ. Even more interesting graphs can be created using the product of theta and a trigonometric function. To get all these elusive points, you put in the r value in both curves to see what additional points you get. The curves and questions Since sin is negative, reflect over \(x\)-axis so it’s on the “bottom”. Jun 15, 2014 - Explore Christina Meng's board "cool graphs..." on Pinterest. Hypnotize your HoneyUse your calculator to graph a spiral that’s sure to catch … With this problem, we can create the following t-chart and see the sequence of petals being drawn. RE: Anybody know any cool looking polar equations? Again, a larger range of values for theta gives more chance for the graph to wrap around. How does the choice of one of these effect the graph? Try creating some other roses on your own with different numbers of petals to verify that the even/odd relationship holds. You can see that they are inscribed in circles of radius 1,2,3,...,12. \(\displaystyle \begin{array}{c}r=2\left( {\frac{y}{r}} \right)+3\left( {\frac{x}{r}} \right)\\{{r}^{2}}=2y+3x\\{{x}^{2}}+{{y}^{2}}=2y+3x\end{array}\)            \(\begin{array}{c}\text{You can complete the square to get circle:}\\{{x}^{2}}-3x+{{y}^{2}}-2y=0\\\left( {{{x}^{2}}-3x+\frac{9}{4}} \right)+\left( {{{y}^{2}}-2y+1} \right)=0+\frac{9}{4}+1\\\underline{{{{{\left( {x-\frac{3}{2}} \right)}}^{2}}+{{{\left( {y-1} \right)}}^{2}}=\frac{{13}}{4}}}\end{array}\). (Positive would be on the right side). Converting from rectangular coordinates to polar coordinates can be a little trickier since we need to check the quadrant of the rectangular point to get the correct angle; the quadrants must match. Here are some examples: The following polar-rectangular relationships are useful in this regard. .leader-1-multi{display:block !important;float:none;line-height:0px;margin-bottom:15px !important;margin-left:0px !important;margin-right:0px !important;margin-top:15px !important;min-height:250px;min-width:970px;text-align:center !important;}eval(ez_write_tag([[300,250],'shelovesmath_com-leader-1','ezslot_4',126,'0','0']));eval(ez_write_tag([[300,250],'shelovesmath_com-leader-1','ezslot_5',126,'0','1']));eval(ez_write_tag([[300,250],'shelovesmath_com-leader-1','ezslot_6',126,'0','2']));I find that drawing polar graphs is a combination of part memorizing and part knowing how to create polar t-charts. .large-mobile-banner-2-multi{display:block !important;float:none;line-height:0px;margin-bottom:15px !important;margin-left:0px !important;margin-right:0px !important;margin-top:15px !important;min-height:250px;min-width:970px;text-align:center !important;}eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-2','ezslot_10',128,'0','0']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-2','ezslot_11',128,'0','1']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-2','ezslot_12',128,'0','2'])); \(r=a\cos \left( {b\theta } \right),\,\,r=a\sin \left( {b\theta } \right)\). |. Think about what values of theta make the sine and cosine maxima! It also hits \(5-3=2\) in the loop. I remember that this line is horizontal since it’s the same as \(r\sin \theta =-4\). These graphs always go through the pole (center). In fact, you’ll walk away thinking that Polar Graphs are super fun and cool to draw because these basic Polar Equations are graphed with the use of plotting-points, symmetry, and the Unit Circle. We already saw that the polar graph is a rose with an odd number of petals (5), each 6 units long, and starts at \(\displaystyle -\frac{{90}}{b}=-\frac{{90}}{5}=-18{}^\circ \). 0 0. In the last section, we learned how to graph a point with polar coordinates (r, θ). Here are some other beautiful botanicals. With Plotly Express, it is possible to represent polar data as scatter markers with px.scatter_polar, and as lines with px.line_polar.. Plotly Express is the easy-to-use, high-level interface to Plotly, which operates on a variety of … }\) In addition, we should always check whether the … Graph of r = 2.5, a limacon. Then the area enclosed by the polar curve is “Neat” Polar Graphs continued page 3 of 4 10) Y= : r1 = 4 cos(4sin(4 cos(4sin(cos(4 sin(cos(4sin( tan(θ))))) r2 = 4 cos(cos(cos( tan(tan(tanθ))))) r3 = r1 + r2 Note: r1 & r2 are not turned on only r3 is turned on Window: θ [0,2π] π/24 X [-7,7] 0 Y [-4,4] 0 11) Y= : r1 = 4 cos(2cosθ)) Window: θ [0,2π] π/24 X [-6,6] 0 Y [ … The graph has 2 petals and the length of each petal is \(a\) (7). Source(s): cool polar equations: https://biturl.im/nlda7. For example, if we wanted to rename the point \(\left( {6,240{}^\circ } \right)\) three other different ways between \(\left[ {-360{}^\circ ,360{}^\circ } \right)\), by looking at the graph above, we’d get \(\left( {-6,60{}^\circ } \right)\)(make \(r\) negative and subtract 180°), \(\left( {6,-120{}^\circ } \right)\) (subtract 360°), and \(\left( {-6,-240{}^\circ } \right)\) (make both negative). Note: I remember that when b is the smallest, it’s a “bean”. Really Cool Graphs. There’s no learning curve – you’ll get a beautiful graph or diagram in minutes, turning raw data into something that’s both visual and easy to understand. Do you recognize the inner shaped of the "single petaled rose"? In order to … Here are graphs that we call Cardioids and Limacons. Note that \(r=-5\) would produce the same graph. Note that we had to add \(\boldsymbol {\pi }\) to our answer since we want Quadrant II. FUGP - Fungraph - Graphs of mathematical functions FUGP - Fungraph - Graphs of mathematical functions- 5 types of graphs:- Single - Piecewise - Parametric - Polar - Multiple - Print and copy graph into clipboard - 15 preset examples - Easy to use - User's manual in PDF format - At home or in the classroom, FunGraph … We also have to be careful since there are “phantom” or “elusive” points, typically at the pole. Since \(b\) (5) is odd, we have \(b\) petals, or 5 petals (we don’t multiply by 2). Enjoy! Here are the formulas: \(\displaystyle r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}\,\,\,\,\,\,\,\,\,\text{(this will be positive)}\), \(\displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)\,\,\,\,\,\,\text{(check for correct quadrant)}\). How do you think about the answers? Lets take a look at all four at once! We will also see phantom points when one of the equations is “\(r=\) constant”, since another way to write this is “\(r=\) the negative of that constant”. As theta increases there is some sort of spiraling effect. With negative cos, they start at the negative positive \(\boldsymbol { x}\)-axis (reflect over \(\boldsymbol { y}\)-axis) and are \(\displaystyle \frac{{360}}{5}\), or 72° apart, going counterclockwise. Note that \(r=-4-4\sin \theta \) would make same graph. Note that \(r=-4+3\sin \theta \) would make same graph. \(\begin{array}{l}r=a+b\cos \theta ,\,\,\,a |b|, and |a| < |b|. The graph above was created with a = ½. r = .1θ and r = θ The general form for a spiral is r = aθ, where θ is the angle measure in radians and a is a number multiplier. Multiplying and Dividing, including GCF and LCM, Powers, Exponents, Radicals (Roots), and Scientific Notation, Introduction to Statistics and Probability, Types of Numbers and Algebraic Properties, Coordinate System and Graphing Lines including Inequalities, Direct, Inverse, Joint and Combined Variation, Introduction to the Graphing Display Calculator (GDC), Systems of Linear Equations and Word Problems, Algebraic Functions, including Domain and Range, Scatter Plots, Correlation, and Regression, Solving Quadratics by Factoring and Completing the Square, Solving Absolute Value Equations and Inequalities, Solving Radical Equations and Inequalities, Advanced Functions: Compositions, Even and Odd, and Extrema, The Matrix and Solving Systems with Matrices, Rational Functions, Equations and Inequalities, Graphing Rational Functions, including Asymptotes, Graphing and Finding Roots of Polynomial Functions, Solving Systems using Reduced Row Echelon Form, Conics: Circles, Parabolas, Ellipses, and Hyperbolas, Linear and Angular Speeds, Area of Sectors, and Length of Arcs, Law of Sines and Cosines, and Areas of Triangles, Introduction to Calculus and Study Guides, Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Rules, Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change, Implicit Differentiation and Related Rates, Differentials, Linear Approximation and Error Propagation, Exponential and Logarithmic Differentiation, Derivatives and Integrals of Inverse Trig Functions, Antiderivatives and Indefinite Integration, including Trig Integration, Riemann Sums and Area by Limit Definition, Applications of Integration: Area and Volume, \(\displaystyle \left( {3,-\frac{{3\pi }}{2}} \right)\), \(\displaystyle \frac{{3\pi }}{2}\)   270, Find the length of each petal, number of petals, spacing between each petal, and the tip of the, \(\displaystyle \theta =-\frac{\pi }{6}\), \(\displaystyle r=\frac{4}{{2+\cos \theta }}\), \({{r}^{2}}\sin \left( {2\theta } \right)=4\), Find the intersection points for the following sets of polar curves (algebraically) and also draw a sketch. For the best answers, search on this site https://shorturl.im/oyrdR. Printable Graph Paper Templates; Large Graph Paper Templates; With the right Paper Template, you don’t have to head out to stores anymore to be disappointed, and you don’t have to buy graph paper in packs of more paper than you realistically need.Here is a stock of free polar graph paper templates that you can … Graphing Polar Equations – Video . Pretty cool, if you will I want to introduce you to the world of polar functions. Circle symmetric with \(y\) axis with diameter 6. Thus, we can discard \(r=0\). You may be asked to rename a point in several different ways, for example, between \(\left[ {-2\pi ,2\pi } \right)\) or \(\left[ {-360{}^\circ ,360{}^\circ } \right)\). Plotly's Python graphing library makes interactive, publication-quality graphs online. We're going to look polar functions of the form f = a sin(n ) and r = a cos(n ) which are sometimes called multi-petaled roses. Note that \(\displaystyle \theta =\frac{{5\pi }}{4}\) and \(\displaystyle \theta =-\frac{{3\pi }}{4}\)  would produce the same graph. The limacon goes out to \(3+5=8\) on the negative \(x\)-axis and hits 3 and –3 on the \(y\)-axis. In the Polar Coordinate System, we go around the origin or the pole a certain distance out, and a certain angle from the positive \(x\)–axis: eval(ez_write_tag([[300,250],'shelovesmath_com-medrectangle-3','ezslot_2',109,'0','0']));The ordered pairs, called polar coordinates, are in the form \(\left( {r,\theta } \right)\), with \(r\) being the number of units from the origin or pole (if \(r>0\)), like a radius of a circle, and \(\theta \) being the angle (in degrees or radians) formed by the ray on the positive \(x\)–axis (polar axis), going counter-clockwise. In polar coordinates, the simplest function for r is r = constant, which makes a circle centered at the origin. Note that when we also get \(r=0\) (the pole) for the answers, this is one point only, and in these cases, the pole is included in the other part of the answers. There are many familiar shapes such as lines, circles, parabolas, and ellipses which can be expressed in polar form. The 1/3 multiplier makes the spiral tighter around the pole. With sin, graph is along \(\displaystyle \frac{\pi }{4}\). Thus, it is typically easier to convert from polar to rectangular. To see all of these varieties in one glance, execute the next block of commands. However, when n is even, the rose has 2n petals. If both \(r\) and the angle \(\theta \) are negative, you have to make sure you go clockwise to get the angle, but in the opposite direction \(r\) units. Return to graph mode. Try it! And now only positive values of R are … Note that you can also use “2nd APPS (ANGLE)” on your graphing calculator to do these conversions, but you won’t get the answers with the roots in them (you’ll get decimals that aren’t “exact”). eval(ez_write_tag([[580,400],'shelovesmath_com-medrectangle-4','ezslot_3',110,'0','0']));For a point \(\left( {r,\theta } \right)\), do you see how you always go counter-clockwise (or clockwise, if you have a negative angle) until you reach the angle you want, and then out from the center \(r\) units, if \(r\) is positive? With positive sin, they start at \(\displaystyle \frac{{90}}{b}=\frac{{90}}{4}=22.5\) degrees from the positive \(\boldsymbol { x}\)-axis (memorize this) and they are \(\displaystyle \frac{{360}}{8}\), or 45° apart, going counterclockwise. When |a| < |b|, the graph not only passes through the origin, but also part of it folds inside itself. Each different color is a different graph. Here are some polar equations that result in lines: Note: There is no “\(r\)” in the equation; just draw line at \(\displaystyle \frac{\pi }{4}\). A chart can represent tabular numeric data, functions or some types of qualitative structure. (Note that since the t-chart starts on the positive \(\boldsymbol { x}\)axis, the \(r\)’s are negative in the chart). I included t -charts in both degrees and radians. Some of the worksheets for this concept are Form meets function polar graphing project, 1 of 2 graphing sine cosine and tangent functions, It is often necessary to transform from rectangular to, Weather and climate work, Note in … Note: I remember that when \(a=b\), things are in harmony, like a heart. }\end{array}\), \(\displaystyle \begin{array}{c}-\sin \theta =\cos \theta ;\,\,\,\,\,\,\tan \theta =-1\\\theta =\frac{{3\pi }}{4}\,\,\left( {r=\cos \left( {\frac{{3\pi }}{4}} \right)=-\frac{{\sqrt{2}}}{2}} \right)\text{ (duplicate)},\,\,\,\,\frac{{7\pi }}{4}\,\,\left( {r=\frac{{\sqrt{2}}}{2}} \right)\\\underline{{\left( {\frac{{\sqrt{2}}}{2},\frac{{7\pi }}{4}} \right),\,\,\left( {0,0} \right)\text{ (”phantom” point)}}}\end{array}\), \(\displaystyle \begin{array}{c}\cos \theta =\cos 2\theta \\\cos \theta =2{{\cos }^{2}}\theta -1\,\,\text{(identity)}\\2{{\cos }^{2}}\theta -\cos \theta -1=0;\,\,\,\,\,\,\left( {2\cos +1} \right)\left( {\cos \theta -1} \right)=0\\\cos \theta =-\frac{1}{2}\,\,\,\,\,\,\,\,\,\,\cos \theta =1\\\theta =\frac{{2\pi }}{3}\,\,\left( {r=\cos \left( {\frac{{2\pi }}{3}} \right)=-\frac{1}{2}} \right),\,\,\frac{{4\pi }}{3}\,\,\left( {r=-\frac{1}{2}} \right),\,\,\,\theta =0\,\,\left( {r=1} \right)\\\underline{{\left( {-\frac{1}{2},\frac{{2\pi }}{3}} \right),\,\,\left( {-\frac{1}{2},\frac{{4\pi }}{3}} \right),\,\,\,\left( {1,0} \right),\,\,\left( {0,0} \right)\text{ (”phantom” point) }}}\end{array}\), \(\displaystyle \begin{array}{c}\sin 2\theta =\cos \theta \\2\sin \theta \cos \theta \,\,\text{(identity)}=\cos \theta \\2\sin \cos \theta -\cos \theta =0;\,\,\,\cos \theta \left( {2\sin \theta -1} \right)=0\\\cos \theta =0\,\,\,\,\,\,\,\,\,\,\sin \theta =\frac{1}{2}\\\theta =\frac{\pi }{2}\,\,\left( {r=\cos \left( {\frac{\pi }{2}} \right)=0} \right),\,\,\,\,\frac{{3\pi }}{2}\,\,\left( {r=0} \right)\,\,\,\text{(duplicate)},\,\\\theta =\frac{\pi }{6}\,\,\left( {r=\frac{{\sqrt{3}}}{2}} \right),\,\,\,\,\frac{{5\pi }}{6}\,\,\left( {r=-\frac{{\sqrt{3}}}{2}} \right)\\\underline{{\left( {0,\frac{\pi }{2}} \right),\,\,\,\left( {\frac{{\sqrt{3}}}{2},\frac{\pi }{6}} \right),\,\,\left( {-\frac{{\sqrt{3}}}{2},\frac{{5\pi }}{6}} \right)}}\end{array}\), \(\displaystyle \frac{{2\pi }}{3}\)    120, \(\displaystyle \frac{{4\pi }}{3}\)    240, \(\displaystyle \frac{{3\pi }}{2}\)    270, \(\displaystyle \frac{\pi }{4}\)        45, \(\displaystyle \frac{\pi }{2}\)        90, \(\displaystyle \frac{{3\pi }}{4}\)     135, \(\displaystyle \frac{{5\pi }}{4}\)     225, \(\displaystyle \frac{{3\pi }}{2}\)     270, \(\displaystyle \frac{{7\pi }}{4}\)     315. Except today, the graphs are going to be pretty and cool. Since it’s negative, reflect over \(x\)-axis so it’s on the “bottom”. \(\begin{array}{c}\text{Use }{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\,\,\text{identity:}\\{{\left( {r\cos \theta } \right)}^{2}}+{{\left( {r\sin \theta } \right)}^{2}}=49\\{{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta =49\end{array}\)                    \(\begin{array}{c}{{r}^{2}}\left( {{{{\cos }}^{2}}\theta +{{{\sin }}^{2}}\theta } \right)=49\\{{r}^{2}}\left( 1 \right)=49\\\underline{{r=\pm 7}}\end{array}\), \(\begin{array}{c}r\sin \theta =-r\cos \theta \\r\sin \theta +r\cos \theta =0\\r\left( {\sin \theta +\cos \theta } \right)=0\end{array}\)                \(\displaystyle \begin{array}{c}\xcancel{{r=0}}\,\,\,\,\,\text{or}\,\,\,\,\sin \theta =-\cos \theta \\\,\tan \theta =-1\\\,\underline{{\theta =\frac{{3\pi }}{4}}}\end{array}\). To find the intersection points for sets of polar curves, it’s helpful to draw the curves and also to solve algebraically. Some of the worksheets displayed are Form meets function polar graphing project, 1 of 2 graphing sine cosine and tangent functions, It is often necessary to transform from rectangular to, Weather and climate work, Note in each section do not … You see spirals in the ocean’s shells and the far-reaches of space. Since twice, \(\displaystyle \begin{array}{c}r\cos \theta =-3\\r=\frac{{-3}}{{\cos \theta }};\,\,\,\,\,\,\,\,\,\,\underline{{r=-3\sec \theta }}\end{array}\), \(\begin{array}{c}r\sin \theta ={{\left( {r\cos \theta } \right)}^{4}}\\r\sin \theta ={{r}^{4}}{{\cos }^{4}}\theta \\r\sin \theta -{{r}^{4}}{{\cos }^{4}}\theta =0\\r\left( {\sin \theta -{{r}^{3}}{{{\cos }}^{4}}\theta } \right)=0\end{array}\)              \(\require {cancel} \displaystyle \begin{array}{c}\xcancel{{r=0}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\sin \theta -{{r}^{3}}{{\cos }^{4}}\theta =0\\\,r=\sqrt[3]{{\frac{{\sin \theta }}{{{{{\cos }}^{4}}\theta }}}}\\\underline{{r=\sqrt[3]{{\tan \theta {{{\sec }}^{3}}\theta }}}}\end{array}\), \(\displaystyle \begin{array}{c}\text{Note that }{{x}^{2}}+{{y}^{2}}={{r}^{2}}:\\{{r}^{2}}=-3\left( {r\cos \theta } \right)\\{{r}^{2}}+3r\cos \theta =0\\r\left( {r+3\cos \theta } \right)=0\end{array}\)              \(\xcancel{{r=0\,}}\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\underline{{r=-3\cos \theta }}\), \(\begin{array}{c}2\left( {r\cos \theta } \right)+r\sin \theta =3\\r\left( {2\cos \theta +\sin \theta } \right)=3\end{array}\)                \(\displaystyle \underline{{r=\frac{3}{{2\cos \theta +\sin \theta }}}}\), \(\displaystyle \begin{array}{c}\sqrt{{{{x}^{2}}+{{y}^{2}}}}=4\left( {\frac{y}{{\sqrt{{{{x}^{2}}+{{y}^{2}}}}}}} \right)\\{{\left( {\sqrt{{{{x}^{2}}+{{y}^{2}}}}} \right)}^{2}}=4y\\{{x}^{2}}+{{y}^{2}}=4y\end{array}\)      \(\begin{array}{c}\text{You can complete the square to get circle:}\\{{x}^{2}}+{{y}^{2}}=4y\\{{x}^{2}}+{{y}^{2}}-4y=0\\{{x}^{2}}+\left( {{{y}^{2}}-4y+4} \right)=0+4\\\underline{{{{x}^{2}}+{{{\left( {y-2} \right)}}^{2}}=4}}\end{array}\), \(\displaystyle {{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)=45{}^\circ ;\,\,\,\,\,\frac{y}{x}=1;\,\,\,\,\,\,\,\,\,\,\,\,\underline{{y=x}}\), \(\displaystyle {{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)=-\frac{\pi }{6};\,\,\,\,\,\frac{y}{x}=-\frac{1}{{\sqrt{3}}};\,\,\,\,\,\,\,\,\,\,\,\,\underline{{y=-\frac{x}{{\sqrt{3}}}\cdot \frac{{\sqrt{3}}}{{\sqrt{3}}}}}=-\frac{{x\sqrt{3}}}{3}\), \(\sqrt{{{{x}^{2}}+{{y}^{2}}}}=5;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{{{{x}^{2}}+{{y}^{2}}=25}}\), \(\displaystyle \begin{array}{c}\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\frac{4}{{2+\frac{x}{{\sqrt{{{{x}^{2}}+{{y}^{2}}}}}}}}\\\sqrt{{{{x}^{2}}+{{y}^{2}}}}\left( {2+\frac{x}{{\sqrt{{{{x}^{2}}+{{y}^{2}}}}}}} \right)=4\\\underline{{2\sqrt{{{{x}^{2}}+{{y}^{2}}}}+x=4}}\end{array}\)              \(\begin{array}{c}\text{It }\!\!’\!\!\text{ s probably easier to leave }r\text{ in the}\\\text{equations and substitute }\sqrt{{{{x}^{2}}+{{y}^{2}}}}\text{ later}\text{. Graphs of Polar Equations . We can also draw circles not centered at the origin. Note: Unlike their rectangular equivalents, \(r=a\pm b\cos \theta \) and \(r=-a\pm b\cos \theta \) (same with \(r=a\pm b\sin \theta \) and \(r=-a\pm b\sin \theta \)) are the same polar graph! Circle symmetric with \(x\)-axis with diameter 4. When n is an odd number, the resulting rose has exactly n petals. To plot a point, you typically circle around the positive \(x\)–axis \(\theta \)  degrees first, and then go out from the origin or pole \(r\) units (if \(r\) is negative, go the other way (180°) \(r\) units). Charts and graphs are visual representations of your data. \(\displaystyle r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{1+25}}=\sqrt{{26}}\), \(\displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{5}{{-1}}} \right)=-1.373+\pi =1.768\text{ (2nd quadrant)}\), \(\displaystyle \left( {\sqrt{{26}},1.768} \right)\).
Dmax S14 Tail Lights, Southern Belle Frozen Foods, Which Of The Following Physical Changes Occur During Puberty?, Range Kleen Non Stick Oven Liner Reviews, National Fire Protection Association Publications, Jacksonville, Fl 32219,