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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 39
Capitolo 5, Problema 39

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)

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Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), where \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
Identify the given polar coordinates: \(r = 7.4\) and \(\theta = 2.5\) radians.
Calculate the \(x\)-coordinate using the formula \(x = 7.4 \times \cos(2.5)\).
Calculate the \(y\)-coordinate using the formula \(y = 7.4 \times \sin(2.5)\).
Express the rectangular coordinates as the ordered pair \((x, y)\) using the values found in the previous steps.

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Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin (radius r) and an angle (θ) measured from the positive x-axis. The given point (7.4, 2.5) means r = 7.4 and θ = 2.5 radians.
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Intro to Polar Coordinates

Conversion from Polar to Rectangular Coordinates

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). This translates the point from a radius-angle format to Cartesian coordinates on the xy-plane.
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Convert Points from Polar to Rectangular

Trigonometric Functions (Sine and Cosine)

Sine and cosine functions relate an angle in a right triangle to the ratios of its sides. Cosine gives the ratio of the adjacent side to the hypotenuse, and sine gives the ratio of the opposite side to the hypotenuse, essential for coordinate conversion.
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Graph of Sine and Cosine Function
Pratica correlata
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In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

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