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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.3.47
Chapter 5, Problem 5.3.47

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

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Recall that to convert rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\).
Calculate the radius \(r\) by substituting \(x = 5\) and \(y = 0\) into the formula: \(r = \sqrt{5^2 + 0^2}\).
Simplify the expression for \(r\) to find the distance from the origin to the point.
Find the angle \(\theta\) by evaluating \(\arctan\left(\frac{0}{5}\right)\), which gives the angle the point makes with the positive \(x\)-axis.
Consider the quadrant where the point \((5, 0)\) lies to determine the correct value of \(\theta\) in radians, remembering that if \(x > 0\) and \(y = 0\), \(\theta\) is either \(0\) or \(2\pi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rectangular Coordinates

Rectangular coordinates (x, y) represent a point's position on a plane using horizontal and vertical distances from the origin. The x-value indicates horizontal displacement, while the y-value indicates vertical displacement. Understanding these coordinates is essential for converting to polar form.
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Convert Points from Polar to Rectangular

Polar Coordinates

Polar coordinates express a point's location using a distance from the origin (r) and an angle (θ) measured from the positive x-axis. The distance r is always non-negative, and θ is typically given in radians. This system is useful for problems involving rotation or circular motion.
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Intro to Polar Coordinates

Conversion Between Rectangular and Polar Coordinates

To convert from rectangular (x, y) to polar (r, θ), calculate r = √(x² + y²) and θ = arctangent(y/x). Special attention is needed when x = 0 or when determining the correct quadrant for θ. Expressing θ in radians ensures consistency in trigonometric calculations.
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Convert Points from Polar to Rectangular
Related Practice
Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 15° + i sin 15°)]³

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Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.


y² = 6x

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (√2 − i)⁴

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 − i)⁵