Problem 23 - Technical Mathematics with Calculus

Question

Write the polar coordinates of each point. $$(3.00,6.00)$$

Step-by-Step Solution

Answer

\( r = 6.71 \), \( \theta = 1.11 \) radians; Polar coordinates: \( (6.71, 1.11) \)

Step 1: Find the radius r

To find the radius r for the polar coordinates, use the distance formula from the origin (0,0) to the point (3.00,6.00), which is given by \( r = \sqrt{x^2 + y^2} \). Substitute x with 3.00 and y with 6.00 to calculate r.

Step 2: Calculate the angle theta

The angle theta (θ) in polar coordinates can be computed using the arctangent function, \( \theta = \arctan(\frac{y}{x}) \). Use the coordinates x = 3.00 and y = 6.00 to find the angle θ in radians (or degrees, if specified).

Step 3: Consider the quadrant for the correct angle

Since the point (3.00, 6.00) is in the first quadrant, where both x and y are positive, the angle \( \theta \) calculated from the arctan function is the correct final angle in polar coordinates.

Key Concepts

Radius Calculation in Polar Coordinates

Calculating the radius is a fundamental step in converting Cartesian coordinates to polar coordinates. The radius is essentially the distance from the origin (0,0) to the point in question. To find this distance, we use the Pythagorean theorem as applied in a two-dimensional coordinate system. Imagine drawing a right triangle where the point's x-coordinate represents one leg, the y-coordinate represents the other leg, and the hypotenuse extends from the origin to the point.

For a point with Cartesian coordinates \( (x, y) \), we calculate the radius \( r \) using the formula \[ r = \sqrt{x^2 + y^2} \]. This formula is derived from the theorem mentioned above. In the given example with the point \( (3.00, 6.00) \), we plug in the x and y values to calculate the radius as follows: \[ r = \sqrt{3.00^2 + 6.00^2} = \sqrt{9 + 36} = \sqrt{45} \]. This result gives us the magnitude of the radius, a crucial component of the polar coordinate system.

Determining the Angle Theta

The angle theta \( (\theta) \) in polar coordinates represents the directional component, indicating how far an angle is rotated from the positive x-axis. To compute \( \theta \) for a point \( (x, y) \), we use the arctangent function, which is the inverse of the tangent trigonometric function.

The formula to calculate \( \theta \) is \[ \theta = \arctan(\frac{y}{x}) \]. It's important to be aware of the quadrant in which the point is located as it affects the angle calculation. The point \( (3.00, 6.00) \) resides in the first quadrant, where both x and y coordinates are positive, making the calculated angle from the arctangent function the correct angle without any adjustments. However, since the arctangent function can only determine angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) (or -90° and 90°), if a point falls in a different quadrant, additional considerations must be taken into account to obtain the actual angle.

Converting Cartesian Coordinates to Polar Coordinates

Polar coordinates provide a different perspective for representing points in a plane using radius and angle, unlike Cartesian coordinates which use x and y values. The conversion process requires both components we discussed: radius and angle theta.

To convert a point from Cartesian to polar coordinates, follow these general steps:

• Calculate the radius as \( r = \sqrt{x^2 + y^2} \) using the square root of the sum of the squares of the x and y coordinates.
• Determine the angle theta \( (\theta) \) using the arctangent function based on the y and x coordinates \( (\arctan(\frac{y}{x}) ) \) and adjust it according to the quadrant where the point lies.
• Combine the radius and angle to express the final polar coordinates as \( (r, \theta) \) where \( r \) is the radius and \( \theta \) is the angle in radians (or degrees).

The example point \( (3.00, 6.00) \) would then be expressed in polar coordinates as \( (r, \theta) \) where \( r \) and \( \theta \) come from the previous calculations. This representation emphasizes the point's distance and direction from the origin, offering a distinctive way to analyze points especially in fields like physics or engineering where radial symmetry is important.