Problem 28 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(-3,150^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right)\).

Step 1: Understanding Polar Coordinates

The point \((-3, 150^{ ext{°}})\) is given in polar coordinates, where \(-3\) is the radius \(r\) and \(150^{ ext{°}}\) is the angle \(\theta\). To convert this into rectangular coordinates \((x, y)\), we will apply the formulas for conversion.

Step 2: Using the Conversion Formulas

The formulas to convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) are:\[ x = r \cdot \cos(\theta) \]\[ y = r \cdot \sin(\theta) \]

Step 3: Calculating the x-coordinate

Substitute \(r = -3\) and \(\theta = 150^{\text{°}}\) into the formula for \(x\):\[ x = -3 \cdot \cos(150^{\text{°}}) \]\(\cos(150^{\text{°}}) = -\cos(30^{\text{°}}) = -\frac{\sqrt{3}}{2}\), so\[ x = -3 \cdot \left(-\frac{\sqrt{3}}{2}\right) = \frac{3\sqrt{3}}{2} \]

Step 4: Calculating the y-coordinate

Substitute \(r = -3\) and \(\theta = 150^{\text{°}}\) into the formula for \(y\):\[ y = -3 \cdot \sin(150^{\text{°}}) \]\(\sin(150^{\text{°}}) = \sin(30^{\text{°}}) = \frac{1}{2}\), so\[ y = -3 \cdot \frac{1}{2} = -\frac{3}{2} \]

Step 5: Conclusion: Rectangular Coordinates

Combining both calculations, the rectangular coordinates are:\(\left(\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a fundamental way of representing points in a two-dimensional plane using the Cartesian coordinate system. They present a point as \(x\) and \(y\) values, each marking a location on the horizontal and vertical axes respectively. This method comes in handy for locating and analyzing points precisely on a plane.

• The \(x\) coordinate tells us the position along the horizontal axis.
• The \(y\) coordinate indicates the position along the vertical axis.

This is often used in conjunction with graphing quadratic functions, trigonometric identities, and in geometry for plotting shapes. Converting to rectangular coordinates can be crucial in understanding graphs and equations in a more straightforward, visual manner.

Polar Coordinates

Polar coordinates define a point in relation to a fixed origin, and use a radius and an angle. Unlike rectangular coordinates, this system specifies a point based on:

• An angle, \(\theta\), which indicates the direction from the origin.
• A radius, \(r\), which measures the distance from the origin.

This system is particularly useful in scenarios involving circular movements, spirals, or when dealing with angles.
To convert polar coordinates to rectangular coordinates, you employ trigonometric transformations. This involves using the angle and radius to ascertain the precise \(x\) and \(y\) coordinates on the Cartesian plane, making understanding and computation more convenient when dealing with non-linear contexts.

Trigonometric Functions

Trigonometric functions are essential for converting between polar and rectangular coordinates. These functions include sine, cosine, and tangent. They see wide use in various fields such as engineering, physics, and navigation.

• Cosine (cos) determines the horizontal distance from the origin.
• Sine (sin) calculates the vertical distance from the origin.

In the exercise, to find the \(x\) coordinate \(x = r \cdot \cos(\theta)\) is used, effectively projecting the radius onto the x-axis. Similarly, for the \(y\) coordinate, \(y = r \cdot \sin(\theta)\) helps ascertain the vertical projection. These functions simplify the transition between polar and rectangular coordinates by exploiting relationships within a right-angled triangle.