Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$\left(\frac{5}{2}, \frac{4}{3}\right)$$

Step-by-Step Solution

Answer

One possible set of polar coordinates for the point \(\left(\frac{5}{2}, \frac{4}{3}\right)\) is \(r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{4}{3}\right)^2}\) and θ approximately equals the numerical value obtained from the graphing utility upon entering 'arctan(8/15)' in degree mode.

Step 1: Compute the radial coordinate r

The radial coordinate r, representing the distance from the origin to the point in the polar plane, is computed using the Pythagorean Theorem as \(r = \sqrt{x^2 + y^2}\). Here, \((x,y) = \left(\frac{5}{2}, \frac{4}{3}\right)\), so, \(r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{4}{3}\right)^2}\).

Step 2: Compute the angular coordinate θ

To find the angular coordinate θ, we need to proceed with caution because simply finding the inverse tangent of y/x can be misleading due to quadrant issues. Only if we are in the first quadrant (both x and y positive), we can use the relation θ= arctan(y/x). Here, \((x,y) = \left(\frac{5}{2}, \frac{4}{3}\right)\), both positive, so that condition is satisfied. Thus, we can compute θ as follows: θ= arctan(4/3 / 5/2) = arctan(8/15). We need to use a graphing utility to find the numerical approximation of this value.

Step 3: Use a graphing utility

To find the numerical approximation of arctan(8/15), use a graphing utility. Make sure the mode in the graphing utility is set to 'degree' mode if the desired answer is in degrees. If the mode is already in 'degree', enter 'arctan(8/15)' to get the approximate value of θ. If it was in 'radian', switch it to degree and repeat

Key Concepts

Graphing Utility

When working with polar coordinates, a graphing utility can be incredibly helpful. It allows you to visualize the relationship between the coordinates on a polar grid. To use a graphing utility effectively:

• Ensure it is set to the correct mode, either 'degree' or 'radian', based on your preference or requirement.
• Enter expressions such as trigonometric functions directly to obtain their numerical values. For instance, calculating \( \text{arctan}(8/15) \) would give you a precise angle.
• Many graphing utilities can convert between rectangular and polar coordinates, simplifying the process of finding multiple coordinate pairs.

By transforming complicated calculations into visual graphs, graphing utilities make understanding and solving mathematical problems much simpler and more intuitive.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental tool in mathematics, particularly when converting between rectangular and polar coordinates. It states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. This principle can be symbolized as:

• Given a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the relation is: \(c^2 = a^2 + b^2\).
• In the context of coordinates, \((x, y) \) becomes \((r, \theta) \), where \(r = \sqrt{x^2 + y^2}\), representing the hypotenuse.

Using the Pythagorean theorem to find the radial coordinate \(r\) ensures that we accurately measure the distance from the origin to the point in the polar plane.

Rectangular Coordinates

Rectangular coordinates are a way of plotting points on a cartesian plane using horizontal and vertical distances, commonly represented as \( (x, y) \). These coordinates are straightforward but may become less intuitive for circular or angular motions. In the exercise, the given point represents the distances:

• \( x = \frac{5}{2} \) — a horizontal distance from the y-axis.
• \( y = \frac{4}{3} \) — a vertical distance from the x-axis.

Despite their simplicity, rectangular coordinates can be easily converted to polar form, allowing us to represent the same location with a radial distance \(r\) and an angle \(\theta\). This conversion highlights different aspects of a point's position, making it useful in different contexts such as navigation and physics.

Inverse Tangent

The inverse tangent, or \( ext{arctan}\), function helps in finding the angle associated with a point when its \((x, y)\) coordinates are known. It is a trigonometric function that computes the angle whose tangent is a given ratio of the opposite to adjacent side.

• When \((x, y)\) are positive, as in the first quadrant, \(\theta = \text{arctan}\left(\frac{y}{x}\right)\) is straightforward.
• The formule \(\theta = \text{arctan}\left(\frac{y}{x}\right)\) enables the conversion from rectangular to polar angles efficiently.
• Employing a graphing utility provides a precise measure of \(\theta\), crucial when the calculation involves non-standard angles or decimals.

However, always verify the quadrant your point lies in before concluding the angle from \( ext{arctan}\) alone, since it might typically output angles between \(-\pi/2\) and \(\pi/2\).