Textbook Question
In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. x⁶ − 1 = 0
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 EquationsProblem 79
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 79Chapter 5, Problem 79
In Exercises 79–80, convert each polar equation to a rectangular equation. Then determine the graph's slope and y-intercept.
r sin (θ − π/4) = 2
Verified step by step guidance1
Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\) and \(\tan \theta = \frac{y}{x}\).
Use the angle difference identity for sine: \(\sin(\theta - \frac{\pi}{4}) = \sin \theta \cos \frac{\pi}{4} - \cos \theta \sin \frac{\pi}{4}\). Substitute this into the given equation to get \(r (\sin \theta \cos \frac{\pi}{4} - \cos \theta \sin \frac{\pi}{4}) = 2\).
Since \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), rewrite the equation as \(r \left( \sin \theta \frac{\sqrt{2}}{2} - \cos \theta \frac{\sqrt{2}}{2} \right) = 2\).
Distribute \(r\) and replace \(r \sin \theta\) with \(y\) and \(r \cos \theta\) with \(x\), yielding \(\frac{\sqrt{2}}{2} y - \frac{\sqrt{2}}{2} x = 2\).
Multiply both sides by \(\sqrt{2}\) to clear the fractions, then rearrange the equation into the slope-intercept form \(y = mx + b\) to identify the slope and y-intercept.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conversion between Polar and Rectangular Coordinates
Polar coordinates (r, θ) relate to rectangular coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Converting a polar equation to rectangular form involves substituting these expressions to rewrite the equation in terms of x and y.
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Convert Points from Polar to Rectangular
Trigonometric Angle Difference Identity
The angle difference identity for sine states that sin(α − β) = sin α cos β − cos α sin β. Applying this identity to sin(θ − π/4) allows the polar equation to be expanded into terms involving sin θ and cos θ, facilitating conversion to rectangular coordinates.
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Slope and Y-Intercept of a Line
Once the equation is in rectangular form (y = mx + b), the slope (m) represents the line's steepness, and the y-intercept (b) is the point where the line crosses the y-axis. Identifying these helps in graphing and understanding the line's behavior.
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Related Practice
Textbook Question
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Textbook Question
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Textbook Question
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