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# MTH 108 Quiz: Chapter 8 Study Guide Notes Premium

9 Pages
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School
University of Miami
Department
Mathematics
Course
MTH 108
Professor
Lun- Yi Tsai
Semester
Fall

Description
MTH 108 Precalculus Mathematics II Chapter 8 Study Guide Notes L. Sterling Abstract Provide a generalization to each of the key terms listed in this section. Pole What is a pole? A pole is a point in the polar coordinate system. Polar Axis What is a polar axis? A polar axis is a ray with a vertex at the pole in the polar coordinate system. Polar and Rectangular Coordinates What are the polar coordinates? Polar Coordinates are ordered pairs in the form of (r;▯). (r;▯) Any point with (r;▯) as their respected polar coordinates, which has ▯ in radians, can represent by either (r;▯ + 2▯k) or even (▯r;▯ + ▯ + 2pik), which notes that k is any integer. On the other hand, (r;▯) are also the poles polar coordinates, which notes that ▯ being any possible angle. What are the rectangular coordinates? Polar Coordinates are ordered pairs in the form of (x;y). Polar to Rectangular Coordinates What are the diﬀerence between polar and rectangular coordinates? P = Point Rectangular Coordinates = Polar=Cartesian Coordinates (x; y) = (r; ▯) x = r cos ▯ y = r sin ▯ 1 Converting Rectangular to Polar Coordinates With Given Point on a Coordinate Axis ▯ What are the steps for converting from rectangular/cartesian to polar coordinates with the given point on a coordinate axis? – Plot the point, which will be labeled (x;y). – Make a note on which quadrant that the point is lying. – Make a note on which coordinate axis that the point is lying. – Find the distance, which is r, from the origin to the given point. – Find ▯. With Given Point on a Quadrant ▯ What are the steps for converting from rectangular/cartesian to polar coordinates with the given point on a quadrant? – Plot the point, which will be labeled (x;y). – Make a note on which quadrant that the point is lying. – Make a note on which coordinate axis that the point is lying. – Find the distance, which is r, from the origin to the given point by using the following: p r = x + y2 ▯ If x = 0, then ﬁnd r. ▯ If y = 0, then ﬁnp r. ▯ If x 6= 0, then r =x + y . p ▯ If y 6= 0, then r =x + y . – Find ▯. ▯ If x = 0, then ﬁnd ▯. ▯ If y = 0, then ﬁnd ▯. ▯ If x 6= 0, then ﬁnd the quadrant the point lies. ▯ If y 6= 0, then ﬁnd the quadrant the point lies. ▯ When either x 6= 0 or y 6= 0, then the following occurs: ▯y▯ Quadrant I : ▯ = tan1 x ▯ y Quadrant II : ▯ = ▯ + tan1 x ▯ ▯ Quadrant III : ▯ = ▯ + tan1 y x ▯1 ▯y▯ Quadrant IV : ▯ = tan x Rectangular to Polar p 2 2 r = x + y r = x + y 2 y tan ▯ = x [x 6= 0] 2 Theorem: Horizontal and Vertical Lines Horizontal Lines a : Real Number Equation : r sin ▯ = a a ▯ 0 : a units lying above the pole a < 0 : jaj units below the pole Vertical Lines a : Real Number Equation : r cos ▯ = a a ▯ 0 : a units lying to the right of the pole a < 0 : jaj units to the left of the pole Theorem: Equations and Circles r = 2a sin ▯ Radius : a Center : (0; a) r = ▯2a sin ▯ Radius : a Center : (0; ▯a) r = 2a cos ▯ Radius : a Center : (a; 0) r = ▯2a cos ▯ Radius : a Center : (▯a; 0) Circles Every circle would be passing through the pole: 3 Theorem: Symmetry Tests Symmetric to X-axis: Polar Axis Replace With ▯ ▯▯ Symmetric to Y -axis: Line Replace With ▯ ▯ ▯ ▯ Symmetric to Origin: Pole Replace With r ▯r Logarithmic Spiral A logarithmic spiral is an equation that can be written like ▯ = n ln(r) as an example with its own spirals while inﬁnite while both away from and towards the pole. Complex Numbers Any numbers that are in the form of z = a + bi where the following occur: ▯ a and b are real numbers ▯ z is the standard form – a is the real part – b is the imaginary part ▯ i is the imaginary unit – So this notes that 2 i = i ▯ i = ▯1 Compositions of Complex Numbers Equality of Complex Numbers ▯ a + bi = c + di – If and only if a = c and b = d Sum of Complex Numbers ▯ (a + bi) + (c + di) = (a + c) + (b + d)i Diﬀerence of Complex Numbers ▯ (a ▯ bi) + (c ▯ di) = (a ▯ c) + (b ▯ d)i 4 Product of Complex Numbers ▯ (a + bi) ▯ (c + di) = (ac ▯ bd) + (ad + bc)i Quotient of Complex Numbers a + bi c + di (a + bi)(c ▯ di) = (c + di)(c ▯ di) (ac ▯ b(▯d)) + (a(▯d) + bc)i = c + d2 ac + bd ▯ adi + bci = 2 2 c + d ac + bd ▯ adi + bci = c + d 2 ac + bd ▯ad + bc = c + d2 + c + d 2 i Conjugates Complex Number and Conjugate ▯ The conjugate of any real number is just the real number. ▯ If z = a + bi is the complex number, then its conjugate, which can be denoted withand deﬁned with the following: z = a + bi = a ▯ bi ▯ The product of a complex number, which is z = a+bi, and its conjugate, which is z = a▯bi is actually a nonnegative real number, which makes the following: (a + bi)(a ▯ bi) = a(a) + a(▯bi) + bi(a) + bi(▯bi) 2 2 2 = a ▯ abi + abi ▯ b i 2 ▯ 2 = a + (1 ▯ 1)abi ▯ b (▯1) 2 2 = a + (01)abi + b = a + b 2 Complex Number’s Product and Conjugates ▯ The conjugate’s conjugate of a complex number is just the origi
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