Problem 54 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (12,-9) $$

Step-by-Step Solution

Answer

The polar coordinates are \((15, 5.64)\).

Step 1: Identify the Rectangular Coordinates

The given rectangular coordinates are \((x, y) = (12, -9)\).

Step 2: Calculate the Radius (r)

The radius \(r\) in the polar coordinate system is determined by the formula \(r = \sqrt{x^2 + y^2}\). Plugging in the values, we have:\[r = \sqrt{12^2 + (-9)^2} = \sqrt{144 + 81} = \sqrt{225} = 15\]

Step 3: Calculate the Angle (θ) in Radian

To find the angle \(\theta\), use the tangent function: \(\tan(\theta) = \frac{y}{x}\).Substituting the given values, we have:\[\tan(\theta) = \frac{-9}{12} = -\frac{3}{4}\]Using a calculator or trigonometric tables to find \(\theta\), note that \(\tan^{-1}(-\frac{3}{4})\) will give you an angle in the fourth quadrant.

Step 4: Adjust θ to Correct Quadrant

The point \((12, -9)\) is in the fourth quadrant where \(\theta\) should be between \(\frac{3\pi}{2}\) and \(2\pi\). The angle \(\tan^{-1}(-\frac{3}{4})\) gives a negative angle, which corresponds to moving clockwise. To find the positive equivalent, add \(2\pi\) or convert by subtracting the absolute value of angle from \(2\pi\). Hence, \[\theta = 2\pi + \tan^{-1}(-\frac{3}{4})\approx 2\pi - 0.6435 \approx 5.64\]

Step 5: Finalize the Polar Coordinates

Combine the values found for \(r\) and \(\theta\) to express the polar coordinates. Thus, the polar coordinates are \((r, \theta) = (15, 5.64)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are an essential way of locating points on a plane. Named after the mathematician René Descartes, this system uses a pair of numerical values to specify a point's position. In the context of two-dimensional space, the coordinates are given as

• \( x \): the horizontal distance from the origin
• \( y \): the vertical distance from the origin

For example, the rectangular coordinates \((12, -9)\) tell us that the point is 12 units to the right (along the x-axis) and 9 units down (along the y-axis) from the origin.
Rectangular coordinates are particularly useful in many mathematical calculations, including finding distances and angles, due to their simplicity and directness in mapping out locations on a grid.

Radius Calculation

A significant step in converting from rectangular to polar coordinates is determining the radius \(r\), which represents the distance of the point from the origin. The formula used, \(r = \sqrt{x^2 + y^2}\), directly stems from the Pythagorean theorem.Given the point \((12, -9)\):

• Square the x-coordinate: \(12^2 = 144\)
• Square the y-coordinate: \((-9)^2 = 81\)
• Add these squares: \(144 + 81 = 225\)
• Find the square root of the sum: \(r = \sqrt{225} = 15\)

By following these steps, we find that the radius \(r\) is 15. This radius helps plot how far away and in which direction the point lies relative to the origin, which is a crucial part of defining its position in polar coordinates.

Angle Calculation

The angle \(\theta\) is a pivotal component in polar coordinates, indicating the direction from the positive x-axis to the point. Calculating this angle involves using trigonometric functions, specifically the tangent function.Using the formula \(\tan(\theta) = \frac{y}{x}\), we substitute the given values:

• \(\tan(\theta) = \frac{-9}{12} = -\frac{3}{4}\)

The calculator or trigonometric tables can provide us with \(\theta = \tan^{-1}(-\frac{3}{4})\). This angle, initially, may appear in the wrong range or sign due to the quadrant of the point. It's necessary to adjust this angle so that it situates correctly between \(0\) and \(2\pi\).When a negative angle is obtained, it implies a clockwise rotation. For the given point's fourth quadrant location, adding \(2\pi\) or subtracting the angle's absolute value gives us a valid \(\theta\) of approximately 5.64 radians.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are at the heart of converting between coordinate systems. These functions translate between the rectangular x and y coordinates and the polar r and \(\theta\).

Tangent links the y and x coordinates as \(\tan(\theta) = \frac{y}{x}\), which is essential for determining the angle of a point.

When working with these functions, it's vital to grasp that they express spatial relationships:

• \(\sin(\theta)\) represents the vertical leg over the hypotenuse.
• \(\cos(\theta)\) denotes the horizontal leg over the hypotenuse.

Understanding these relationships empowers us to move fluently between representing points in different coordinate systems. Moreover, becoming comfortable with functions like \(\tan^{-1}\) is crucial for finding the correct angles, often involving quadrant-specific adjustments to ensure the solution fits the standard range of angles \(0 \leq \theta < 2\pi\).