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Chapter 6: Problem 34

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. . (5.4,2.85)

Short Answer

Expert verified
For the given polar coordinates (5.4, 2.85), after calculation, the rectangular coordinates will be (x,y). Note: values of x and y will be rounded values.

Step by step solution

01

Identify the Polar Coordinates

The provided polar coordinates are (5.4, 2.85). We can identify \( r = 5.4 \) and \( \theta = 2.85 \) radians.
02

Conversion to Rectangular Coordinates

In order to convert the polar coordinates into rectangular coordinates, we will use two basic trigonometric relationships. The x-coordinate can be obtained by multiplying the radial coordinate r by \( cos(\theta) \). Similarly, the y-coordinate is obtained by multiplying r by \( sin(\theta) \). Using these formulas, the rectangular coordinates (x,y) will be \( x = rcos(\theta) \) and \( y=rsin(\theta) \).
03

Calculate the X-coordinate (round to 2 decimal places)

From the formula derived in Step 2, we will calculate the x-coordinate using the given polar coordinates. It will be \( x = 5.4 * cos(2.85) \). Use a calculator to find this value.
04

Calculate the Y-coordinate (round to 2 decimal places)

Similarly, we will calculate the y-coordinate using the given polar coordinates. It will be \( y = 5.4 * sin(2.85) \). Use a calculator to find this value.
05

State the Final Coordinates

Now, with the x and y coordinates calculated, the final rectangular coordinates can be stated.

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Key Concepts

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

polar coordinates
Polar coordinates represent a point in a plane using a distance from a reference point and an angle from a reference direction. The reference point is called the pole, which is similar to the origin in rectangular coordinates, and the reference direction is the positive x-axis. A point in polar coordinates is given as
  • \( (r, \theta) \), where \( r \) is the radius or the distance from the pole,
  • and \( \theta \) is the angle measured from the positive x-axis.
This system is particularly useful in situations where motion naturally occurs in a circular path or direction, such as rotating particles or planets. The given polar coordinates \( (5.4, 2.85) \) indicate a point located 5.4 units from the pole, forming an angle of 2.85 radians with the positive x-axis.
rectangular coordinates
Rectangular coordinates are more commonly known as Cartesian coordinates. These coordinates specify the position of a point by defining its distances along two perpendicular axes, usually labeled as the x-axis and y-axis. A point in rectangular coordinates is given as:
  • \( (x, y) \), where \( x \) is the horizontal distance from the origin,
  • and \( y \) is the vertical distance from the origin.
This system is widely used in various applications, including graphs and maps, as it allows the easy plotting and identification of points, lines, and shapes. Conversion from polar coordinates to rectangular coordinates involves calculating these x and y values from the polar radius and angle, using trigonometric functions.
trigonometric relationships
To convert polar coordinates to rectangular coordinates, we rely on fundamental trigonometric relationships. The sine and cosine functions help find the x and y coordinates from polar coordinates. Here鈥檚 how this works:
  • The x-coordinate: \( x = r \cos(\theta) \)
  • The y-coordinate: \( y = r \sin(\theta) \)
Here, \( r \) and \( \theta \) are given from the polar coordinates. The cosine function affects the horizontal distance (x), while the sine function influences the vertical distance (y). These trigonometric calculations are crucial for transforming a point from the polar to the rectangular system, aiding in various calculations and applications in physics and engineering.
radians
Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 parts, radians provide a direct relationship between the circumference of a circle and its radius.
  • A full circle is \( 2\pi \) radians.
  • A half-circle, or straight line angle, is \( \pi \) radians.
  • Therefore, 1 radian equals the angle created when the arc length is equivalent to the radius of the circle.
Using radians in trigonometric functions often simplifies calculations in equations or expressions, especially in calculus. For the given example, the polar angle \( 2.85 \) radians implies the angular displacement from the positive x-axis and is used to compute the corresponding x and y coordinates.

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Most popular questions from this chapter

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