Problem 24 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(-5,-\frac{9 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left( -\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2} \right)\).

Step 1: Understand the Problem

We want to convert the point given in polar coordinates \((-5, -\frac{9\pi}{4})\) to rectangular coordinates \((x, y)\). Polar coordinates represent points as \((r, \theta)\), where \(r\) is the radius, and \(\theta\) is the angle measured in radians.

Step 2: Use Conversion Formulas

To convert from polar to rectangular coordinates, we use the formulas: \[ x = r \cos \theta \]\[ y = r \sin \theta \].Substitute \(r = -5\) and \(\theta = -\frac{9\pi}{4}\) into these formulas.

Step 3: Simplify the Angle

The angle \(-\frac{9\pi}{4}\) is not within the standard range \([0, 2\pi)\). We add \(2\pi\) until it falls within the standard range. Since \(-\frac{9\pi}{4} + 2\pi = -\frac{\pi}{4}\), use this equivalent angle for calculating.

Step 4: Calculate the Rectangular Coordinates

Now, calculate the rectangular coordinates using the standard angle:\[ x = -5 \cos\left(-\frac{\pi}{4}\right) = -5 \cdot \frac{\sqrt{2}}{2} = -\frac{5\sqrt{2}}{2} \]\[ y = -5 \sin\left(-\frac{\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \].

Step 5: Write the Result

Putting it together, the rectangular coordinates are \(\left( -\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2} \right)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with angles and their relationships in geometric problems. Some of the fundamental identities include the sine, cosine, and tangent functions, which relate the angles of a triangle to the ratios of its sides. In the context of polar and rectangular coordinates, the sine and cosine functions are vital for converting between these systems. When we have a point in polar coordinates \(r, \theta\), the rectangular coordinates can be found using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). This is because:

• \(\cos \theta\) represents the horizontal proportion of the radius \(r\), which gives us the x-coordinate.
• \(\sin \theta\) represents the vertical proportion of the radius \(r\), which gives us the y-coordinate.

Understanding these identities helps us convert the location and direction from one coordinate system to another seamlessly. This shows how interconnected trigonometric principles are with geometry and algebra in real-world applications.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles, and it’s often necessary to convert between them in various math problems. In most scientific and mathematical contexts, radians are used, because they are directly related to the properties of circles. The entire circle is \(2\pi\) radians or \(360^\circ\). To convert radians into degrees, you can multiply the radian measure by \(\frac{180}{\pi}\). Conversely, to turn degrees into radians, multiply the degree measure by \(\frac{\pi}{180}\). For instance, the angle \(-\frac{9\pi}{4}\) in radians can be converted to degrees as follows:

• \( -\frac{9\pi}{4} \times \frac{180}{\pi} = -405^\circ \)

It's useful to note these conversions, as they ensure accurate interpretations and operations involving angles across different systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a 2D space based on its distance from two perpendicular axes, usually labeled as x and y. This coordinate system is widely used because it directly represents the position as a straightforward grid of perpendicular lines. In this system:

• \(x\)-axis goes left to right, and \(y\)-axis goes bottom to top.
• A point is written as \((x, y)\).

This form is intuitive for analyzing properties such as distances between points or slopes of lines. In the context of converting from polar to rectangular coordinates, you derive the \(x\) and \(y\) values based on the radius and angle measurements from the polar format. The benefit of using rectangular coordinates is the ease with which points can be plotted and the simplicity of understanding geometric relationships and algebraic equations.

Polar Coordinates

Polar coordinates offer a different way to locate points in a 2D plane. Unlike rectangular coordinates, polar coordinates \(r, \theta\) describe a point by its radial distance from a fixed origin and the angle from a polar axis, typically the positive \(x\)-axis, measured in radians.Key features of polar coordinates include:

• \(r\): the distance from the origin. It can be positive, negative, or zero.
• \(\theta\): the angle made with the positive \(x\)-axis. It can be a positive or negative measure, offering different directions.

This system is especially useful for problems involving circular motion or spirals and is favored in fields like physics and engineering for its ability to effectively represent circular and rotational systems. If we take the point given in the exercise, \(-5, -\frac{9\pi}{4}\), we translate it into a rectangular system, allowing further analysis and computations on a more common grid layout.