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In Exercises 1–3, perform the indicated operations and write the result in standard form. ___ ___ 2√−49 + 3√−64
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 3
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 3Capitolo 5, Problema 3
In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, 5π/4)
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Recall that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis.
Note that the given point is \((-3, \frac{5\pi}{4})\). Since \(r\) is negative, the point lies in the direction opposite to the angle \(\frac{5\pi}{4}\).
Find the angle directly opposite to \(\frac{5\pi}{4}\) by adding or subtracting \(\pi\) (180 degrees): calculate \(\theta_{opposite} = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\).
Plot the point by moving a distance of \(3\) units (the absolute value of \(r\)) from the origin in the direction of \(\theta_{opposite} = \frac{\pi}{4}\).
Compare this location with points A, B, C, and D on the graph to determine which one corresponds to the coordinates \((-3, \frac{5\pi}{4})\).
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Polar Coordinates
Polar coordinates represent a point in the plane using a distance from the origin (radius) and an angle measured from the positive x-axis. The format is (r, θ), where r is the radius and θ is the angle in radians or degrees.
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05:32
Intro to Polar Coordinates
Negative Radius in Polar Coordinates
A negative radius means the point is located in the direction opposite to the angle θ. To plot (−r, θ), you move r units in the direction θ + π (180 degrees), effectively reversing the direction.
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05:32Intro to Polar Coordinates
Angle Measurement in Radians
Angles in polar coordinates are often given in radians, where 2π radians equal 360 degrees. Understanding how to convert and interpret angles like 5π/4 helps locate the correct direction on the graph.
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5:04Converting between Degrees & Radians
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In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 7 − 4t, y = 5 + 6t; t = 1
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In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = t² + 1, y = 5 − t³; t = 2
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In Exercises 1–8, add or subtract as indicated and write the result in standard form. (3 + 2i) − (5 − 7i)
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In Exercises 1–3, perform the indicated operations and write the result in standard form. 5 / 2−i
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In Exercises 1–10, plot each complex number and find its absolute value. z = 4i
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