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Chapter 7: Problem 3

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(4, \frac{\pi}{2}\right)\)

Short Answer

Expert verified
The rectangular coordinates of the given polar coordinates \(\left(4, \frac{\pi}{2}\right)\) are \((x, y) = (0, 4)\).

Step by step solution

01

Identify the polar coordinates

Let's identify the polar coordinates provided: \(r = 4\) \(\theta = \frac{\pi}{2}\)
02

Apply the conversion formula for the x-coordinate

Now, let's apply the conversion formula for the x-coordinate: \(x = r\cos(\theta) = 4\cos\left(\frac{\pi}{2}\right)\)
03

Evaluate the x-coordinate

We can evaluate the x-coordinate expression: \(x = 4\cos\left(\frac{\pi}{2}\right) = 4(0) = 0\)
04

Apply the conversion formula for the y-coordinate

Now, let's apply the conversion formula for the y-coordinate: \(y = r\sin(\theta) = 4\sin\left(\frac{\pi}{2}\right)\)
05

Evaluate the y-coordinate

We can evaluate the y-coordinate expression: \(y = 4\sin\left(\frac{\pi}{2}\right) = 4(1) = 4\)
06

Write the rectangular coordinates

With both the x and y values evaluated, we can now write the rectangular coordinates of the point: \((x, y) = (0, 4)\)

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Key Concepts

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

Polar Coordinates
Polar coordinates are a way of representing points in a plane using a combination of a distance and an angle. This method is quite different from the usual Cartesian system we might be used to. Instead of using two perpendicular axes with values for x and y, polar coordinates use:
  • **A radius** \(r\), which is the distance from a fixed point called the origin.
  • **An angle** \(\theta\), which is typically measured in radians from a fixed direction.
These two values \((r, \theta)\) can represent any point in the plane. It is especially useful in scenarios where the relationship is best described in terms of radius and angle, such as circular or spiral patterns.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent points using two values, x and y, which correspond to horizontal and vertical distances from the origin. This system is one of the most common methods used in mathematics due to its straightforward grid-like representation.
  • The **x-coordinate** indicates the horizontal position.
  • The **y-coordinate** indicates the vertical position.
Every point is a combination of these two values, written as \((x, y)\). In the context of converting from polar to rectangular coordinates, the aim is to find corresponding x and y values for a given polar point. This involves using trigonometric functions to adjust values based on radius and angle.
Trigonometric Functions
Trigonometric functions are essential components for converting polar coordinates to rectangular coordinates. These functions help bridge the gap between polar angle-distance representation and Cartesian x-y representation. The two main functions used in this process are:
  • **Sine** \(\sin(\theta)\): helps determine the vertical component of the point.
  • **Cosine** \(\cos(\theta)\): helps determine the horizontal component of the point.
For a point described by \((r, \theta)\), the formulas to convert to rectangular coordinates are:
  • \(x = r\cos(\theta)\): provides the horizontal component.
  • \(y = r\sin(\theta)\): provides the vertical component.
Using these functions ensures that the conversion from the radial form to the grid-like form is both accurate and intuitive.
Coordinate Systems
Coordinate systems like polar and rectangular are organizational schemas for representing the position of a point. Each system has its distinct advantages depending on the geometric or real-world problem at hand.
  • **Rectangular systems** are great for linear motion and regular, grid-based layouts. They're ideal for everyday mapping and spatial design.
  • **Polar systems** cater to radial and angular dynamics, such as those found in rotational mechanics or navigation on a circular path.
Understanding how to navigate between these systems, through trigonometric conversions, widens the scope and flexibility of mathematical modeling and problem-solving. The key is recognizing which system most naturally aligns with the problem, and how to skillfully perform conversions when needed.

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Most popular questions from this chapter

$$ \text { Find the magnitude of the vector }(-5,-2) \text { . } $$

$$ \text { Find the magnitude of the vector }(-3,2) $$

Find real numbers \(a\) and \(b\) such that \(2+3 i\) and \(2-3 i\) are roots of the polynomial \(x^{2}+a x+b\).

Write each expression in the form \(a+b i,\) where a and b are real numbers. \((6+5 i)^{2}\)

Suppose $$ f(x)=a x^{2}+b x+c $$ where \(a \neq 0\) and \(b^{2}<4 a c .\) Verify by direct substitution into the formula above that $$ f\left(\frac{-b+\sqrt{4 a c-b^{2}} i}{2 a}\right)=0 $$ and $$ f\left(\frac{-b-\sqrt{4 a c-b^{2}} i}{2 a}\right)=0 $$
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