Problem 28 - Technical Mathematics with Calculus

Question

Write the polar coordinates of each point. $$(207,186)$$

Step-by-Step Solution

Answer

\(r = 279.57\) (rounded to two decimal places), \(\theta = 42.01^\circ\) (rounded to two decimal places).

Step 1: Calculate the radius r

The radius r can be found using the Pythagorean theorem, where r is the hypotenuse of a right-angled triangle with base x and height y. Use the formula: \(r = \sqrt{x^2 + y^2}\). Substitute x = 207 and y = 186 to calculate r.

Step 2: Calculate the angle θ

To find the angle θ, use the arctangent function with y as the opposite side and x as the adjacent side. Apply the formula: \(\theta = \arctan(\frac{y}{x})\), and use x = 207 and y = 186. Ensure that the angle is in the correct quadrant or make necessary adjustments based on the signs of x and y.

Step 3: Convert the angle to degrees

If needed and preferred, convert the angle θ from radians to degrees using the conversion: \(\text{degrees} = \theta \times \frac{180}{\pi}\). Otherwise, leave it expressed in radians.

Key Concepts

The Pythagorean Theorem

When it comes to finding the distance from the origin to a point in the Cartesian coordinate system, the Pythagorean theorem is essential. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This mathematical principle is represented by the equation: \( r = \sqrt{x^2 + y^2} \), where \(x\) and \(y\) are the lengths of the sides of the triangle and \(r\) is the hypotenuse. In our exercise, the Cartesian coordinates (207, 186) can be viewed as defining a right-angled triangle with 207 and 186 being the lengths of the legs adjacent to the right angle and the hypotenuse being the distance from the origin to the point — which is what we're trying to find as the radius in polar coordinates.

The Arctangent Function

Once we've determined the radius, our next step in the conversion to polar coordinates involves finding the angle \(\theta\). The angle can be found using the arctangent function, often denoted as \(\arctan\) or \(\tan^{-1}\).

This trigonometric function is the inverse of the tangent function and is used to find an angle when the opposite and adjacent sides of a right-angled triangle are known. The formula for this is \(\theta = \arctan(\frac{y}{x})\), where \(x\) and \(y\) are the Cartesian coordinates of the point. For instance, in our problem, we would calculate the angle as \(\arctan(\frac{186}{207})\).

It's crucial to consider the quadrant in which our point lies because the arctangent function can only determine the angle with respect to the positive x-axis and may give an angle in the wrong quadrant if not adjusted for properly.

Angle Conversion

Angles can be measured in both degrees and radians, and converting between them is a common task in mathematics and engineering. Radians are often used in calculus and higher maths due to their natural connection with the arc lengths on a circle, while degrees are usually preferred for their interpretability and their basis in geometry.

The conversion from radians to degrees relies on the equivalency that 180 degrees is equal to \(\pi\) radians. The conversion formula looks like this: \(\text{degrees} = \theta \times \frac{180}{\pi}\). On the flip side, to convert from degrees to radians, you use \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\). In our example involved in the step-by-step solution, using the angle found from the arctangent function, we can easily convert it to degrees, if the solution requires that notation.