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Chapter 10: Problem 8

Plot the point with these polar coordinates. $$\left[\frac{1}{3} , \frac{2}{3} \pi\right]$$

Short Answer

Expert verified
The point with polar coordinates \(\left(\frac{1}{3} , \frac{2}{3} \pi\right)\) can be converted to Cartesian coordinates which are approximately \(\left(-\frac{1}{6}, \frac{\sqrt{3}}{6}\right)\). Plot this point on the Cartesian plane by locating the intersection of the x value -1/6 and the y value \(\frac{\sqrt{3}}{6}\), and label the point accordingly.

Step by step solution

01

Convert polar coordinates to Cartesian coordinates

To convert polar coordinates \((r,\theta)\) to Cartesian coordinates \((x,y)\), we use the following equations: \[ x = r\cos \theta \] and \[ y = r\sin \theta \] In our case, the polar coordinates are \(\left(\frac{1}{3} , \frac{2}{3} \pi\right)\). So, we can find the Cartesian coordinates (x, y) by substituting the values of r and 胃 in the equations above: \[ x = \frac{1}{3} \cos \left(\frac{2}{3} \pi\right) \] and \[ y = \frac{1}{3} \sin \left(\frac{2}{3} \pi\right) \]
02

Calculate x and y values

Now we can calculate the x and y values: \[ x = \frac{1}{3} \cos \left(\frac{2}{3} \pi\right) \approx -\frac{1}{6} \] and \[ y = \frac{1}{3} \sin \left(\frac{2}{3} \pi\right) \approx \frac{\sqrt{3}}{6} \] So the Cartesian coordinates for the given polar coordinates are approximately \(\left(-\frac{1}{6}, \frac{\sqrt{3}}{6}\right)\).
03

Plot the point on the Cartesian plane

Finally, we can plot the point \(\left(-\frac{1}{6}, \frac{\sqrt{3}}{6}\right)\) on the Cartesian plane. 1. Locate the negative x value, which is -1/6 on the x-axis. 2. Then locate the positive y value, which is approximately \(\frac{\sqrt{3}}{6}\) on the y-axis. 3. Draw a point at the intersection of those two locations, completing the plot with the necessary label. Now the point with the given polar coordinates \(\left(\frac{1}{3} , \frac{2}{3} \pi\right)\) is successfully plotted on the Cartesian plane.

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Key Concepts

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

Cartesian Coordinates
Cartesian coordinates are a fundamental concept in mathematics used to describe the position of a point in a plane. Named after the French mathematician Ren茅 Descartes, the system relies on two perpendicular lines called axes鈥攖he x-axis (horizontal) and the y-axis (vertical). The intersection of these axes is known as the origin, designated by the point (0,0). Each point's location is determined by an ordered pair (x, y), representing the distances along the x-axis and the y-axis respectively.
  • The first number in the pair, x, reflects the point's horizontal distance from the origin. Positive values indicate a position to the right of the origin, while negative values indicate a position to the left.
  • The second number, y, indicates the vertical distance from the origin. Positive values are above the origin, while negative values are below it.
Understanding Cartesian coordinates is crucial as they form the basis for graphing equations and functions, allowing us to visually interpret mathematical relationships.
Coordinate Conversion
Coordinate conversion is an important process when working with different systems, such as converting between polar and Cartesian coordinates. This can be particularly useful in fields like physics and engineering, where different coordinate systems simplify different problems. Polar coordinates (r, 胃) specify the position of a point based on its distance from the origin (r) and the angle (胃) formed with the positive x-axis.
To convert polar coordinates to Cartesian coordinates, you'll use trigonometric functions:
  • x-coordinate: Use the formula \( x = r\cos \theta \) to determine the x-value by multiplying the radius by the cosine of the angle.
  • y-coordinate: Use the formula \( y = r\sin \theta \) to find the y-value by multiplying the radius by the sine of the angle.
By understanding these conversion formulas, you can transform any coordinate point from polar to Cartesian form, enabling the point to be plotted or utilized within standard graphing techniques.
Trigonometric Functions
Trigonometric functions are mathematical tools that relate the angles of a triangle to the lengths of its sides. They are vital not just in the study of geometry, but also in converting between polar and Cartesian coordinates. The primary trigonometric functions are sine, cosine, and tangent.
  • Cosine (cos 胃): For a given angle 胃, cosine is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
  • Sine (sin 胃): Sine is the ratio of the opposite side to the hypotenuse for angle 胃 in a right triangle.
  • Both sine and cosine functions are periodic, meaning they repeat their values in a cyclical pattern. This periodic nature is fundamental when calculating positions on a circular path, such as those described in polar coordinates.
By applying sine and cosine functions in coordinate conversion, you can efficiently determine the Cartesian coordinates from the given polar coordinates, utilizing elements of both linear and circular measurement.

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Most popular questions from this chapter

Sketch the curves and find the points at which they intersect. Express your answers in rectangular coordinates. $$r=1-\cos \theta, \quad r=1+\sin \theta$$

Give a parametrization for the upper half of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) that satisfies the assumptions made for Exercises \(33-36\)

(a) Let \(a>0 .\) Find the length of the path traced out by $$\begin{array}{l}x(\theta) \quad 3 a \cos \theta+a \cos ^{2} \theta \\\y(\theta)=3 a \sin \theta-a \sin 3 \theta\end{array}$$ as \(\theta\) ranges from 0 to \(2 \pi\). (b) Show that this path can also be parametrized by $$x(\theta)=4 a \cos ^{3} \theta, \quad y(\theta)=4 a \sin ^{3} \theta \quad 0 \leq \theta \cdot 2 \pi$$

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