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Chapter 7: Problem 9

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$\theta=\frac{\pi}{4}$$

Short Answer

Expert verified
The graph of the polar equation \(\theta=\frac{\pi}{4}\) is a line that passes through the origin and makes an angle of 45 degrees with the polar axis.

Step by step solution

01

Understanding the Polar Equation

The given polar equation is \(\theta=\frac{\pi}{4}\), where \(\theta\) is the angle from the positive x-axis. Since \(\theta\) is constant, all points lying on this graph will make an angle of \(\frac{\pi}{4}\) radians (or 45 degrees) with the positive x-axis.
02

Plot the Angle

The next step is to draw the angle \(\theta=\frac{\pi}{4}\) on the polar coordinate plane. Start from the positive x-axis (also called the polar axis) and measure an angle of \(\frac{\pi}{4}\) radians counter-clockwise.
03

Sketch the Graph

Draw a line that passes through the origin and makes an angle of \(\frac{\pi}{4}\) with the polar axis. As \(\theta\) is constantly \(\frac{\pi}{4}\) and r, the distance from origin, can be any real number, this line includes all points that fulfill the polar equation and is therefore the graph of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Graph
Polar graphs represent points in a plane using polar coordinates, which differ fundamentally from the Cartesian coordinate system. In a polar graph, each point is defined by a radial distance from a central point (usually the origin) and an angle from a reference direction, typically the positive x-axis, also known as the polar axis.
  • Structure: Unlike the rectangular grid format of Cartesian coordinates, polar graphs are typically circular, with concentric circles indicating the distance from the origin.
  • Complex Graphs: Polar graphs are versatile as they can easily portray circular and spiral patterns that are complicated to express in Cartesian coordinates.
  • Graph Representation: An equation in polar form such as \ heta = \frac{\pi}{4}\ indicates that the angle is fixed, resulting in a straight line emanating from the origin.

For the given equation \(\theta = \frac{\pi}{4}\), the resulting graph is a line where all points share the same angle measured from the positive x-axis, expanding outward in all radial directions.
Angle Measurement
Angles in polar coordinates are measured from the polar axis, moving counter-clockwise. Proper angle measurement is crucial when working with polar graphs because it determines the position of points and lines.
  • Reference Direction: Starting from the positive x-axis, angles are measured counterclockwise. This direction is standard unless specified otherwise.
  • Degrees and Radians: Angles can be expressed in degrees or radians. With radians, you receive a more mathematical and scale-free approach.
  • Understanding \(\frac{\pi}{4}\): In terms of degrees, \(\frac{\pi}{4}\) translates to 45 degrees, making it a common benchmark angle often appearing in polar equations.

The concept of measuring an angle in polar coordinates centers around visualizing this turning motion from a fixed line to determine the direction of the points in the graph.
Radians
Radians are the standard unit of angular measurement used in mathematics, especially trigonometry and calculus. They provide a natural way to express angles, any angle in a circle can be uniquely represented using radians.
  • Definition: A radian is defined as the angle made at the center of a circle by an arc whose length is equal to the circle's radius.
  • Conversion Factor: There is a straightforward conversion between degrees and radians: 180 degrees is equal to \(\pi\) radians, so \(1\) degree equals \(\frac{\pi}{180}\) radians.
  • Practical Use: Using radians simplifies many mathematical equations, as it ties directly into the arc length, with easier integration and differentiation in calculus contexts.

Understanding radians, especially in the context of polar coordinates, aids in analyzing angles like \(\frac{\pi}{4}\) and integrating them seamlessly into mathematical representations and graphing.

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Most popular questions from this chapter

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=\left\langle\frac{1}{4}, \frac{1}{2}\right\rangle, \mathbf{v}=\left\langle-\frac{1}{2}, \frac{3}{4}\right\rangle$$

Round your answers to two decimal places. The net velocity of a ship is the vector sum of the velocity imparted to the ship by its engine and the velocity of the wind. The engine propels the ship at a velocity of 20 miles per hour in the direction \(S 35^{\circ} \mathrm{E}\). (a) What are the components of the velocity imparted to the ship by its engine? (b) If the wind is blowing from north to south at 12 miles per hour, find the magnitude and direction of the net velocity of the ship. (c) Rework part (b) for the case where the wind is blowing from north to south at 15 miles per hour.

Geometry Marisa has a triangular sign made with her last name on it. She has the sign attached to her lamppost so that visitors can easily identify her house. The lengths of two edges of the sign are 10 inches and 7 inches, and the angle opposite the 10 -inch edge is \(75^{\circ} .\) What is the length of the third edge?

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(10 ;\) direction \(190^{\circ}\)

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why you cannot use the Law of Sines to solve an oblique triangle if you are given only the three sides of the triangle (SSS) and no two of them are of equal length.
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