Problem 48 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$ (0,-6) $$

Step-by-Step Solution

Answer

The polar coordinates of the point (0,-6) are \((6, \frac{3\pi}{2})\).

Step 1: Calculate the radius \(r\)

Polar coordinates are typically written in the form \((r,\theta)\), where \(r\) is the distance from the origin to the point and \(\theta\) is the angle from the positive x-axis to the point. The radius \(r\) is found using the Pythagorean theorem \(r = \sqrt{x^{2} + y^{2}} = \sqrt{0^2 + (-6)^2} = 6\).

Step 2: Calculate the angle \(\theta\)

The angle \(\theta\) can be found by using the tangent function, \(\theta = atan2(y, x)\), but in this case, the function gives \(0\) because x is \(0\). We can see that the point lies on the negative y-axis, so \(\theta = \frac{3\pi}{2}\).

Step 3: Write the polar coordinates

The polar coordinates are therefore \((r, \theta) = (6, \frac{3\pi}{2})\).

Key Concepts

Rectangular to Polar Conversion

Understanding the conversion from rectangular to polar coordinates is essential for navigating different mathematical contexts, such as complex numbers, vector fields, and calculus in polar form. It means transforming the x and y coordinates of a point in the rectangular (or Cartesian) coordinate system to a radius and an angle in the polar coordinate system. The radius, denoted as \( r \), represents the distance from the point to the origin (0,0). To calculate \( r \), you apply the Pythagorean theorem to the coordinates of the point. Once you have \( r \), the next step is to find the angle \( \theta \), which is the counterclockwise angle from the positive x-axis to the point.

When performing this conversion, it's critical to pay attention to the signs of the rectangular coordinates to determine the correct angle. In quadrants II and III, where x is negative, the computed angle from the inverse tangent function should be added to \( \pi \) radians to get the true angle. However, in the case of coordinates (0, -6), since it lies directly on the negative y-axis, the angle can be directly determined as \( \frac{3\pi}{2} \) radians without using the inverse tangent function.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that provides the relationship between the lengths of the sides of a right triangle. It states that 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. In formula terms, if \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse, then the theorem is expressed as \( a^2 + b^2 = c^2 \).

For converting between rectangular and polar coordinates, you use a similar approach since the radius \( r \) in polar coordinates acts like the hypotenuse of a right triangle, with the leg lengths being the x and y coordinates. So, if you have a point with coordinates \( (x, y) \), the distance \( r \) from the origin to that point is given by \( r = \sqrt{x^2 + y^2} \). This allows you to translate a two-dimensional point from its flat Cartesian coordinates into the r and \( \theta \) of polar coordinates effectively.

Angle in Radians

Angles can be measured in degrees or radians, but radians are often used in higher mathematics for their natural connection to the properties of circles and the trigonometric functions. One complete revolution around a circle is \( 2\pi \) radians, which is equivalent to 360 degrees. To convert degrees to radians, you multiply by \( \frac{\pi}{180} \).

Radians are especially useful when dealing with polar coordinates and arc lengths because the formulas integrate seamlessly. For instance, when a point's polar coordinates include an angle in radians, this value directly relates to the arc measurement on the unit circle, making calculations in calculus much simpler. Recognizing the angle in radians in specific cases, like when a point lies on an axis, can also simplify the conversion process, as seen with the point (0, -6) which directly yields the angle \( \frac{3\pi}{2} \) radians.

Atan2 Function

The atan2 function is an extension of the tangential inverse function that helps determine the angle \( \theta \) in polar coordinates, taking into account the signs of both the x and y coordinates, thus placing the angle in the correct quadrant. Unlike the basic arctan function, which only returns values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), atan2 returns values in the full range of \( -\pi \) to \( \pi \), which corresponds to angles of \( -180^\circ \) to \( 180^\circ \).

The function is generally used in the format atan2(y, x), where y and x are the coordinates of a point. However, for points that lie directly on the axes, like (0, -6), the function will not provide the correct angle due to x being zero. In such a scenario, knowing the inherent geometry of the coordinate system allows for immediate determination of the angle without the need to rely entirely on the atan2 function.