[latex]\\tan\\theta = \\frac{y}{x}[\/latex], then use [latex]\\theta = \\tan^{-1}(\\frac{y}{x})[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nBe careful with the angle [latex]\\theta[\/latex]. Check which quadrant the point [latex](x, y)[\/latex] is in, as inverse tangent only gives values in quadrants I and IV. Adjust [latex]\\theta[\/latex] accordingly.<\/p>\r\n
Transforming Equations Between Forms<\/strong><\/p>\r\n\r\n\r\n \t- [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]<\/li>\r\n \t
- [latex]r^2 = x^2 + y^2[\/latex]<\/li>\r\n \t
- [latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\r\n<\/ul>\r\n
\r\n \t- For rectangular to polar: substitute [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex], simplify, and solve for [latex]r[\/latex] in terms of [latex]\\theta[\/latex].<\/li>\r\n \t
- For polar to rectangular: replace [latex]r\\cos\\theta[\/latex] with [latex]x[\/latex] and [latex]r\\sin\\theta[\/latex] with [latex]y[\/latex], use [latex]r^2 = x^2 + y^2[\/latex] when needed, then simplify.<\/li>\r\n<\/ul>\r\n
Graphing in Polar Coordinates<\/h3>\r\n\r\n \t- Symmetry about [latex]\\theta = \\frac{\\pi}{2}[\/latex] (y-axis)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](-r, -\\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\r\n \t
- Symmetry about polar axis (x-axis)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](r, -\\theta)[\/latex] or [latex](-r, \\pi - \\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\r\n \t
- Symmetry about the pole (origin)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](-r, \\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\r\n<\/ul>\r\n
Finding Key Features<\/strong>:<\/p>\r\n\r\n\r\n \t- Zeros<\/strong>: Set [latex]r = 0[\/latex] and solve for [latex]\\theta[\/latex]<\/li>\r\n \t
- Maximum [latex]|r|[\/latex]<\/strong>: Find when the trig function reaches its maximum value (1 for sine\/cosine)<\/li>\r\n \t
- Intercepts<\/strong>: Evaluate [latex]r[\/latex] at [latex]\\theta = 0, \\frac{\\pi}{2}, \\pi, \\frac{3\\pi}{2}[\/latex]<\/li>\r\n<\/ul>\r\n
Classic Polar Curves<\/strong>:<\/p>\r\nCircles<\/strong>:<\/p>\r\n\r\n\r\n \t- [latex]r = a\\cos\\theta[\/latex]: circle with diameter [latex]|a|[\/latex], centered at [latex](\\frac{a}{2}, 0)[\/latex]<\/li>\r\n \t
- [latex]r = a\\sin\\theta[\/latex]: circle with diameter [latex]|a|[\/latex], centered at [latex](0, \\frac{a}{2})[\/latex]<\/li>\r\n<\/ul>\r\n
Cardioids<\/strong> (heart-shaped):<\/p>\r\n\r\n\r\n \t- Forms: [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex] where [latex]\\frac{a}{b} = 1[\/latex] ([latex]a > 0, b > 0[\/latex])<\/li>\r\n \t
- Passes through the pole<\/li>\r\n<\/ul>\r\n
Lima\u00e7ons<\/strong> (snail-shaped):<\/p>\r\n\r\n\r\n \t- Forms: [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex]<\/li>\r\n \t
- Dimpled<\/strong> (one loop with dimple): [latex]1 < \\frac{a}{b} < 2[\/latex]<\/li>\r\n \t
- Convex<\/strong> (no dimple): [latex]\\frac{a}{b} \\geq 2[\/latex]<\/li>\r\n \t
- Inner-loop<\/strong>: [latex]a < b[\/latex] (creates a loop inside)<\/li>\r\n<\/ul>\r\n
Lemniscates<\/strong> (figure-eight or infinity symbol):<\/p>\r\n\r\n\r\n \t- Forms: [latex]r^2 = a^2\\cos(2\\theta)[\/latex] or [latex]r^2 = a^2\\sin(2\\theta)[\/latex]<\/li>\r\n \t
- Always symmetric with respect to the pole<\/li>\r\n \t
- Cosine version is also symmetric about polar axis and [latex]\\theta = \\frac{\\pi}{2}[\/latex]<\/li>\r\n<\/ul>\r\n
Rose Curves<\/strong> (flower petals):<\/p>\r\n\r\n\r\n \t- Forms: [latex]r = a\\cos(n\\theta)[\/latex] or [latex]r = a\\sin(n\\theta)[\/latex]<\/li>\r\n \t
- If [latex]n[\/latex] is even<\/strong>: curve has [latex]2n[\/latex] petals<\/li>\r\n \t
- If [latex]n[\/latex] is odd<\/strong>: curve has [latex]n[\/latex] petals<\/li>\r\n \t
- Each petal has length [latex]|a|[\/latex]<\/li>\r\n<\/ul>\r\n
Archimedes' Spiral<\/strong>:<\/p>\r\n\r\n\r\n \t- Form: [latex]r = \\theta[\/latex] (for [latex]\\theta \\geq 0[\/latex])<\/li>\r\n \t
- Distance from pole increases at constant rate as the curve spirals outward<\/li>\r\n<\/ul>\r\n
Polar Form of Complex Numbers<\/h3>\r\n\r\n \t- The complex plane plots complex number [latex]z = a + bi[\/latex] with real part [latex]a[\/latex] on the horizontal axis and imaginary part [latex]b[\/latex] on the vertical axis.<\/li>\r\n \t
- Absolute Value (Modulus): [latex]|z| = \\sqrt{a^2 + b^2}[\/latex]<\/li>\r\n \t
- Polar Form<\/strong>: [latex]z = r(\\cos\\theta + i\\sin\\theta) = r\\text{ cis }\\theta[\/latex]. Where [latex]r = |z| = \\sqrt{a^2 + b^2}[\/latex] (the modulus) and [latex]\\theta[\/latex] is the argument (angle from positive real axis). We have [latex]\\cos\\theta = \\frac{a}{r}[\/latex] and [latex]\\sin\\theta = \\frac{b}{r}[\/latex].\r\n
\r\n \t- Converting Rectangular to Polar: Find [latex]r = \\sqrt{a^2 + b^2}[\/latex], find [latex]\\theta[\/latex] using [latex]\\cos\\theta = \\frac{a}{r}[\/latex] or [latex]\\tan\\theta = \\frac{b}{a}[\/latex] (watch the quadrant), then write as [latex]r(\\cos\\theta + i\\sin\\theta)[\/latex].<\/span><\/li>\r\n \t
- Converting Polar to Rectangular: Evaluate [latex]\\cos\\theta[\/latex] and [latex]\\sin\\theta[\/latex], calculate [latex]a = r\\cos\\theta[\/latex] and [latex]b = r\\sin\\theta[\/latex], then write as [latex]a + bi[\/latex].<\/span><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
- Operations in Polar Form:<\/span>\r\n
\r\n \t- Product - Multiply moduli, add angles: [latex]z_1 z_2 = r_1 r_2[\\cos(\\theta_1 + \\theta_2) + i\\sin(\\theta_1 + \\theta_2)][\/latex]<\/span><\/li>\r\n \t
- Quotient - Divide moduli, subtract angles: [latex]\\frac{z_1}{z_2} = \\frac{r_1}{r_2}[\\cos(\\theta_1 - \\theta_2) + i\\sin(\\theta_1 - \\theta_2)][\/latex]<\/span><\/li>\r\n \t
- De Moivre's Theorem (for powers): [latex]z^n = r^n[\\cos(n\\theta) + i\\sin(n\\theta)][\/latex]. <\/span>Raise the modulus to the [latex]n[\/latex]th power and multiply the angle by [latex]n[\/latex].<\/li>\r\n \t
- nth Root Theorem (for roots): [latex]z^{\\frac{1}{n}} = r^{\\frac{1}{n}}\\left[\\cos\\left(\\frac{\\theta + 2k\\pi}{n}\\right) + i\\sin\\left(\\frac{\\theta + 2k\\pi}{n}\\right)\\right][\/latex]. <\/span>Where [latex]k = 0, 1, 2, ..., n-1[\/latex] gives all [latex]n[\/latex] distinct roots. Add [latex]\\frac{2k\\pi}{n}[\/latex] to [latex]\\frac{\\theta}{n}[\/latex] to obtain each root.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
Key Equations<\/h2>\r\n
\r\n\r\n\r\nPolar to rectangular<\/strong><\/td>\r\n| [latex]x = r\\cos\\theta[\/latex]\r\n\r\n[latex]y = r\\sin\\theta[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nRectangular to polar\u00a0<\/strong><\/td>\r\n| [latex]r = \\sqrt{x^2 + y^2}[\/latex]\r\n\r\n[latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nCircle (polar form)<\/strong><\/td>\r\n| [latex]r = a\\cos\\theta[\/latex]\r\n\r\n[latex]r = a\\sin\\theta[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nCardioid<\/strong><\/td>\r\n| [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex]\r\nwhere [latex]\\frac{a}{b} = 1[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nLima\u00e7on<\/strong><\/td>\r\n| [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex]\r\nwhere [latex]\\frac{a}{b} \\neq 1[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nLemniscate<\/strong><\/td>\r\n| [latex]r^2 = a^2\\cos(2\\theta)[\/latex] or [latex]r^2 = a^2\\sin(2\\theta)[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nRose curve<\/strong><\/td>\r\n| [latex]r = a\\cos(n\\theta)[\/latex] or [latex]r = a\\sin(n\\theta)[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nArchimedes' spiral<\/strong><\/td>\r\n| [latex]r = \\theta[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nComplex number absolute value<\/strong><\/td>\r\n| [latex]|z| = \\sqrt{a^2 + b^2}[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nComplex number polar form<\/strong><\/td>\r\n| [latex]z = r(\\cos\\theta + i\\sin\\theta) = r\\text{ cis }\\theta[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nProduct of complex numbers<\/strong><\/td>\r\n| [latex]z_1 z_2 = r_1 r_2\\text{ cis}(\\theta_1 + \\theta_2)[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nQuotient of complex numbers<\/strong><\/td>\r\n| [latex]\\frac{z_1}{z_2} = \\frac{r_1}{r_2}\\text{ cis}(\\theta_1 - \\theta_2)[\/latex]<\/td>\r\n<\/tr>\r\n | \r\nDe Moivre's Theorem<\/strong><\/td>\r\n| [latex]z^n = r^n[\\cos(n\\theta) + i\\sin(n\\theta)][\/latex]<\/td>\r\n<\/tr>\r\n | \r\nnth Root Theorem<\/strong><\/td>\r\n[latex]z^{\\frac{1}{n}} = r^{\\frac{1}{n}}\\left[\\cos\\left(\\frac{\\theta + 2k\\pi}{n}\\right) + i\\sin\\left(\\frac{\\theta + 2k\\pi}{n}\\right)\\right][\/latex]\r\nwhere [latex]k = 0, 1, 2, ..., n-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nGlossary<\/h2>\r\nabsolute value of a complex number<\/strong><\/p>\r\nThe distance from the origin to the point representing the complex number in the complex plane; for [latex]z = a + bi[\/latex], it equals [latex]\\sqrt{a^2 + b^2}[\/latex].<\/p>\r\n Archimedes' spiral<\/strong><\/p>\r\nA polar curve with equation [latex]r = \\theta[\/latex] that spirals outward from the pole with distance increasing at a constant rate.<\/p>\r\n argument<\/strong><\/p>\r\nThe angle [latex]\\theta[\/latex] in the polar form of a complex number, measured from the positive real axis.<\/p>\r\n cardioid<\/strong><\/p>\r\nA heart-shaped polar curve with equation [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex] where [latex]\\frac{a}{b} = 1[\/latex].<\/p>\r\n complex plane<\/strong><\/p>\r\nA coordinate plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers; used to plot complex numbers.<\/p>\r\n De Moivre's Theorem<\/strong><\/p>\r\nA formula for raising a complex number in polar form to a power: [latex]z^n = r^n[\\cos(n\\theta) + i\\sin(n\\theta)][\/latex].<\/p>\r\n lemniscate<\/strong><\/p>\r\nA figure-eight shaped polar curve with equation [latex]r^2 = a^2\\cos(2\\theta)[\/latex] or [latex]r^2 = a^2\\sin(2\\theta)[\/latex].<\/p>\r\n lima\u00e7on<\/strong><\/p>\r\nA snail-shaped family of polar curves with equation [latex]r = a \\pm b\\cos\\theta[\/latex] or [latex]r = a \\pm b\\sin\\theta[\/latex]; includes dimpled, convex, and inner-loop varieties depending on the ratio [latex]\\frac{a}{b}[\/latex].<\/p>\r\n modulus<\/strong><\/p>\r\nThe absolute value or magnitude of a complex number; the distance [latex]r[\/latex] from the origin in polar form.<\/p>\r\n polar axis<\/strong><\/p>\r\nThe reference direction in the polar coordinate system, corresponding to the positive x-axis; the starting line from which angles are measured.<\/p>\r\n polar coordinates<\/strong><\/p>\r\nAn ordered pair [latex](r, \\theta)[\/latex] where [latex]r[\/latex] is the distance from the pole and [latex]\\theta[\/latex] is the angle from the polar axis.<\/p>\r\n polar form of a complex number<\/strong><\/p>\r\nThe representation [latex]z = r(\\cos\\theta + i\\sin\\theta)[\/latex] or [latex]r\\text{ cis }\\theta[\/latex], where [latex]r[\/latex] is the modulus and [latex]\\theta[\/latex] is the argument.<\/p>\r\n pole<\/strong><\/p>\r\nThe origin of the polar coordinate system, corresponding to the point [latex](0, 0)[\/latex] in rectangular coordinates.<\/p>\r\n rose curve<\/strong><\/p>\r\nA petal-shaped polar curve with equation [latex]r = a\\cos(n\\theta)[\/latex] or [latex]r = a\\sin(n\\theta)[\/latex]; has [latex]n[\/latex] petals if [latex]n[\/latex] is odd, [latex]2n[\/latex] petals if [latex]n[\/latex] is even.<\/p>\r\n symmetry with respect to the polar axis<\/strong><\/p>\r\nA graph is symmetric about the x-axis if replacing [latex](r, \\theta)[\/latex] with [latex](r, -\\theta)[\/latex] yields an equivalent equation.<\/p>\r\n symmetry with respect to the pole<\/strong><\/p>\r\nA graph is symmetric about the origin if replacing [latex](r, \\theta)[\/latex] with [latex](-r, \\theta)[\/latex] yields an equivalent equation.<\/p>\r\n symmetry with respect to [latex]\\theta = \\frac{\\pi}{2}[\/latex]<\/strong><\/p>\r\nA graph is symmetric about the y-axis if replacing [latex](r, \\theta)[\/latex] with [latex](-r, -\\theta)[\/latex] yields an equivalent equation.<\/p>","rendered":" Essential Concepts<\/h2>\nPolar Coordinates<\/h3>\nUnlike the rectangular (Cartesian) coordinate system that uses perpendicular axes and coordinates [latex](x, y)[\/latex], the polar coordinate system<\/strong> locates points using a distance from the origin and an angle from a fixed direction.<\/p>\n\n- Components of the Polar System:<\/li>\n<\/ul>\n
\n- \n
\n- The pole<\/strong>: the origin point, corresponding to [latex](0, 0)[\/latex] in rectangular coordinates<\/li>\n
- The polar axis<\/strong>: the reference direction (like the positive x-axis)<\/li>\n
- Distance [latex]r[\/latex]<\/strong>: the radial distance from the pole<\/li>\n
- Angle [latex]\\theta[\/latex]<\/strong>: the direction measured counterclockwise from the polar axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
\n- Plotting Polar Points [latex](r, \\theta)[\/latex]: Start at the pole, rotate counterclockwise by angle [latex]\\theta[\/latex] from the polar axis, then move distance [latex]r[\/latex] in that direction. When [latex]r < 0[\/latex], move in the opposite direction (rotate by [latex]\\pi[\/latex] and use [latex]|r|[\/latex]).<\/li>\n
- Multiple Representations: A single point has infinitely many polar representations because angles can be coterminal ([latex](r, \\theta) = (r, \\theta + 2\\pi k)[\/latex]) and negative radius reverses direction ([latex](r, \\theta) = (-r, \\theta + \\pi)[\/latex]).<\/span><\/li>\n
- Converting Polar to Rectangular – Given [latex](r, \\theta)[\/latex], find [latex](x, y)[\/latex]:<\/span><\/li>\n<\/ul>\n
\n- \n
\n- [latex]x = r\\cos\\theta[\/latex]<\/li>\n
- [latex]y = r\\sin\\theta[\/latex]<\/li>\n<\/ul>\n<\/li>\n
- Converting Rectangular to Polar – Given [latex](x, y)[\/latex], find [latex](r, \\theta)[\/latex]:\n
\n- [latex]r = \\sqrt{x^2 + y^2}[\/latex] (always take positive [latex]r[\/latex] when converting)<\/li>\n
- [latex]\\tan\\theta = \\frac{y}{x}[\/latex], then use [latex]\\theta = \\tan^{-1}(\\frac{y}{x})[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
Be careful with the angle [latex]\\theta[\/latex]. Check which quadrant the point [latex](x, y)[\/latex] is in, as inverse tangent only gives values in quadrants I and IV. Adjust [latex]\\theta[\/latex] accordingly.<\/p>\n Transforming Equations Between Forms<\/strong><\/p>\n\n- [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex]<\/li>\n
- [latex]r^2 = x^2 + y^2[\/latex]<\/li>\n
- [latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\n<\/ul>\n
\n- For rectangular to polar: substitute [latex]x = r\\cos\\theta[\/latex] and [latex]y = r\\sin\\theta[\/latex], simplify, and solve for [latex]r[\/latex] in terms of [latex]\\theta[\/latex].<\/li>\n
- For polar to rectangular: replace [latex]r\\cos\\theta[\/latex] with [latex]x[\/latex] and [latex]r\\sin\\theta[\/latex] with [latex]y[\/latex], use [latex]r^2 = x^2 + y^2[\/latex] when needed, then simplify.<\/li>\n<\/ul>\n
Graphing in Polar Coordinates<\/h3>\n\n- Symmetry about [latex]\\theta = \\frac{\\pi}{2}[\/latex] (y-axis)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](-r, -\\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\n
- Symmetry about polar axis (x-axis)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](r, -\\theta)[\/latex] or [latex](-r, \\pi - \\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\n
- Symmetry about the pole (origin)<\/strong>: Replace [latex](r, \\theta)[\/latex] with [latex](-r, \\theta)[\/latex]. If equivalent, the graph has this symmetry.<\/li>\n<\/ul>\n
Finding Key Features<\/strong>:<\/p>\n\n- Zeros<\/strong>: Set [latex]r = 0[\/latex] and solve for [latex]\\theta[\/latex]<\/li>\n
- Maximum [latex]|r|[\/latex]<\/strong>: Find when the trig function reaches its maximum value (1 for sine\/cosine)<\/li>\n
- Intercepts<\/strong>: Evaluate [latex]r[\/latex] at [latex]\\theta = 0, \\frac{\\pi}{2}, \\pi, \\frac{3\\pi}{2}[\/latex]<\/li>\n<\/ul>\n
Classic Polar Curves<\/strong>:<\/p>\nCircles<\/strong>:<\/p>\n\n- [latex]r = a\\cos\\theta[\/latex]: circle with diameter [latex]|a|[\/latex], centered at [latex](\\frac{a}{2}, 0)[\/latex]<\/li>\n
- [latex]r = a\\sin\\theta[\/latex]: circle with diameter [latex]|a|[\/latex], centered at [latex](0, \\frac{a}{2})[\/latex]<\/li>\n<\/ul>\n
Cardioids<\/strong> (heart-shaped):<\/p>\n | | | | | | | | | | | | | | |