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 33

Chapter 5, Problem 33

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)

Verified step by step guidance

Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), which is given by the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Identify the given polar coordinates: \(r = 4\) and \(\theta = 90^\circ\).

Convert the angle \(\theta\) from degrees to radians if necessary, but since trigonometric functions can be evaluated directly in degrees, you can use \(90^\circ\) as is.

Calculate the rectangular coordinates using the formulas: \(x = 4 \cos(90^\circ)\) and \(y = 4 \sin(90^\circ)\).

Evaluate the cosine and sine values for \(90^\circ\) and substitute them back into the expressions for \(x\) and \(y\) to find the rectangular coordinates.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

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 degrees or radians.

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 and angle to Cartesian x and y values.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions is essential, as they relate angles to ratios of sides in right triangles. Also, knowing how to work with angles in degrees or radians ensures correct evaluation of these functions during conversion.

Recommended video:

6:04

Introduction to Trigonometric Functions

Related Practice

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1

555

views

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3

699

views

Textbook Question

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 20(cos 205° + i sin 205°)

509

views

Textbook Question

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 = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

501

views

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of 8i

558

views

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)

548

views