Problem 23 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(9, \frac{7 \pi}{2}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (0, -9).

Step 1: Recall the Formulas

To convert from polar to rectangular coordinates, use the formulas:\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]where \(r\) is the radius and \(\theta\) is the angle in radians.

Step 2: Identify the Polar Coordinates

In the given polar coordinates \((9, \frac{7\pi}{2})\), \(r = 9\) and \(\theta = \frac{7\pi}{2}\).

Step 3: Normalize the Angle

The angle \(\theta = \frac{7\pi}{2}\) is larger than \(2\pi\). To find an equivalent angle within the range \([0, 2\pi]\), subtract \(2\pi\) until the angle is within range. \[ \frac{7\pi}{2} - 2\pi = \frac{7\pi}{2} - \frac{4\pi}{2} = \frac{3\pi}{2} \]Thus, the equivalent angle is \(\frac{3\pi}{2}\).

Step 4: Calculate the Rectangular Coordinates

Substitute \(r = 9\) and \(\theta = \frac{3\pi}{2}\) into the formulas:For \(x\):\[ x = 9 \cos\left(\frac{3\pi}{2}\right) = 9 \times 0 = 0 \]For \(y\):\[ y = 9 \sin\left(\frac{3\pi}{2}\right) = 9 \times (-1) = -9 \]Hence, the rectangular coordinates are \((0, -9)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to describe a point in a plane using two numbers. These numbers represent:

• x: the horizontal distance from the origin.
• y: the vertical distance from the origin.

In this system, the location of a point is easily visualized on a grid laid out in both the horizontal and vertical directions. Rectangular coordinates provide a straightforward way to perform calculations and graphically represent points or shapes.
Consider a point with rectangular coordinates \((x, y) \), which tells us the exact position on the plane. If we take the point \((0, -9) \) as an example, it would be located right along the y-axis, moving 9 units downward from the origin at \(0, 0\). This visual and numerical aspect allows for a clear understanding and manipulation of these coordinates in different mathematical contexts.

Trigonometric Functions

Trigonometric functions are central to converting polar coordinates to rectangular coordinates. These functions relate the angles in a right triangle to the ratios of its sides and are defined as:

• Cosine (cos): adjacent side over hypotenuse.
• Sine (sin): opposite side over hypotenuse.

For an angle \(\theta\), using trigonometric functions, the conversion to rectangular coordinates occurs through the formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

Here, \( r \) is the radius, which denotes the distance from the origin to the point in polar form. The trigonometric function \(*\)\ translates this distance into the \(*\)-direction in the Cartesian system.
For instance, when we have an angle like \(\frac{3\pi}{2}\), the cosine remains 0, and the sine becomes -1, leading to coordinates changing based solely on these values after plugging into the radius in these expressions.

Angle Normalization

Angle normalization ensures angles remain within a standard range, typically between \(0\) and \(2\pi\) radians. Angles wider than one complete rotation (over \(2\pi\) radians) or negative angles need adjustments to fit within this limit, making calculations simpler and results clearer.
For example, the angle given as \(\frac{7\pi}{2}\) is more than \(2\pi\). By subtracting \(2\pi\) sequentially from \(\frac{7\pi}{2}\) to normalize it, the result is \(\frac{3\pi}{2}\). This normalized angle allows direct use in trigonometric calculations without further misconceptions regarding its positioning in the plane. This approach facilitates working with angles consistently in mathematical operations, ensuring they always represent a real expected position in one geometric cycle.

Coordinate Systems

Coordinate systems form the backbone of geometry, providing frameworks for locating positions in space. The two main systems are:

• Polar Coordinates: Define a point based on its distance from the origin (radius \(r\)) and angle \(\theta\) from the positive x-axis.
• Rectangular Coordinates: Represent points with two numbers reflecting positions on an x-y grid.

Polar coordinates are useful in problems dealing with circular or spherical objects, offering an intuitive way of describing points in relation to a central point. However, rectangular coordinates are often preferred for calculations involving straight lines or grid-based layouts.
To convert polar coordinates to rectangular, it requires understanding both systems: knowing your radial distance and angle with trigonometric complements translates that point in terms of a grid structure. The ability to shift between these systems provides versatility in handling various mathematical and real-world spatial problems effectively.