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Chapter 8: Problem 12

Convert the given polar coordinates to Cartesian coordinates. $$ (7,1) $$

Short Answer

Expert verified
The Cartesian coordinates are approximately (3.7821, 5.8905).

Step by step solution

01

Identify Polar Coordinates

The polar coordinates given are \((r, \theta) = (7, 1)\), where \(r\) is the radius and \(\theta\) is the angle in radians.
02

Apply Conversion Formulas

To convert from polar to Cartesian coordinates, use the formulas: \(x = r \cdot \cos(\theta)\)\(y = r \cdot \sin(\theta)\).
03

Calculate the x-coordinate

Substitute the values of \(r\) and \(\theta\) into the formula for \(x\):\(x = 7 \cdot \cos(1)\). Using a calculator to find \(\cos(1)\) in radians gives approximately 0.5403.Thus, \(x \approx 7 \cdot 0.5403 = 3.7821\).
04

Calculate the y-coordinate

Substitute the values of \(r\) and \(\theta\) into the formula for \(y\):\(y = 7 \cdot \sin(1)\). Using a calculator to find \(\sin(1)\) in radians gives approximately 0.8415.Thus, \(y \approx 7 \cdot 0.8415 = 5.8905\).
05

State the Cartesian Coordinates

The Cartesian coordinates corresponding to the polar coordinates \((7, 1)\) are approximately \((x, y) \approx (3.7821, 5.8905)\).

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Key Concepts

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

Polar Coordinates
Polar coordinates provide a unique way of describing the position of a point in a plane. Instead of using horizontal and vertical distances like in the Cartesian system, polar coordinates use:
  • The distance from the point to a fixed origin, known as the radius ( ").
  • An angle ( ">\(\theta\)) measured from a fixed direction, typically from the positive x-axis.
A polar coordinate is generally denoted as \(r, \theta\). In the exercise, the given polar coordinates are \(7, 1\), meaning:
  • \( r = 7 \) is the distance from the origin.
  • \( \theta = 1 \) is the angle in radians.
Cartesian Coordinates
Cartesian coordinates are used to specify the exact location of a point on a two-dimensional plane using two perpendicular lines called axes:
  • The x-axis is the horizontal line.
  • The y-axis is the vertical line.
With Cartesian coordinates, a point is described by the coordinates \(x, y\) representing:
  • The horizontal position (x).
  • The vertical position (y).
The task was to convert the polar coordinates (7, 1) into Cartesian coordinates using trigonometric functions to find the equivalent x and y.
Trigonometric Functions
Trigonometric functions are a crucial part of transforming polar coordinates to Cartesian coordinates. They relate angles to side ratios in right-angled triangles, and their key functions include:
  • Cosine ( ">\(\cos(\theta)\)): Gives the ratio of the adjacent side over the hypotenuse of a right triangle.
  • Sine ( ">\(\sin(\theta)\)): Gives the ratio of the opposite side over the hypotenuse.
To convert polar coordinates \(r, \theta\) into Cartesian form \(x, y\), use these equations:
  • \( x = r \cdot \cos(\theta) \)
  • \( y = r \cdot \sin(\theta) \)
Plugging the polar values \(r = 7, \theta = 1\) into these formulas gives the Cartesian coordinates \( x \approx 3.7821 \) and \( y \approx 5.8905 \).
Radians
Radians are the standard unit of angular measure in mathematics. Unlike degrees, radians provide a natural way to measure angles based on the radius of a circle. The core idea of radians is that the angle is the length of the arc created by that angle when drawn on the unit circle.
  • One complete rotation around a circle is \(2\pi\) radians.
  • Hence, \( \pi \) radians equal 180 degrees.
In our example, the angle \(\theta = 1\) is already given in radians. Using radians in trigonometric functions streamlines calculations, especially in calculus and advanced math topics. This measure facilitates the direct use of functions like \(\cos\) and \(\sin\) without converting from degrees.

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