Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 5

Chapter 5, Problem 5

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 = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

Verified step by step guidance

Identify the given parametric equations: \(x = 4 + 2 \cos t\) and \(y = 3 + 5 \sin t\).

Substitute the given parameter value \(t = \frac{\pi}{2}\) into the expression for \(x\): calculate \(x = 4 + 2 \cos \left( \frac{\pi}{2} \right)\).

Substitute the same parameter value \(t = \frac{\pi}{2}\) into the expression for \(y\): calculate \(y = 3 + 5 \sin \left( \frac{\pi}{2} \right)\).

Evaluate the trigonometric functions \(\cos \left( \frac{\pi}{2} \right)\) and \(\sin \left( \frac{\pi}{2} \right)\) using known unit circle values.

Combine the results from the previous steps to find the coordinates \((x, y)\) of the point on the curve corresponding to \(t = \frac{\pi}{2}\).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.

Evaluating Trigonometric Functions at Specific Angles

To find coordinates for a given parameter t, you substitute t into the trigonometric functions (cos t and sin t). Knowing exact values of sine and cosine at common angles like π/2 is essential for accurate calculation of points on the curve.

Coordinate Calculation from Parametric Form

Once the parameter value is substituted, calculate x and y by performing arithmetic operations on the trigonometric results. This yields the precise point (x, y) on the plane curve corresponding to the given t.

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Related Practice

Textbook Question

In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.

A hyperbola: x = h + a sec t, y = k + b tan t

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Textbook Question

In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)

Textbook Question

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 8

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(3√(−5) )( −4√(−12) )

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Textbook Question

Convert each rectangular equation to a polar equation that expresses r in terms of θ.

y = 3

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Textbook Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Circle: Center: (3,5); Radius: 6

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