Problem 19 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(11,-\frac{7 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left( -\frac{11\sqrt{3}}{2}, -\frac{11}{2} \right)\).

Step 1: Understand Polar Coordinates

In polar coordinates, a point is represented by \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. The given point is \((11, -\frac{7\pi}{6})\) where \(r = 11\) and \(\theta = -\frac{7\pi}{6}\).

Step 2: Apply Formulas for Conversion

To convert from polar to rectangular coordinates, use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] Substitute \(r = 11\) and \(\theta = -\frac{7\pi}{6}\) into these formulas.

Step 3: Calculate the Rectangular Coordinate 'x'

Using the formula \(x = r \cos(\theta)\), substitute the values:\[ x = 11 \cos\left(-\frac{7\pi}{6}\right) \]The cosine of \(-\frac{7\pi}{6}\) is the same as the cosine of \(\pi - \frac{\pi}{6}\), which is \(-\sqrt{3}/2\). Thus,\[ x = 11 \times \left(-\frac{\sqrt{3}}{2}\right) = -\frac{11\sqrt{3}}{2} \]

Step 4: Calculate the Rectangular Coordinate 'y'

Using the formula \(y = r \sin(\theta)\), substitute the values:\[ y = 11 \sin\left(-\frac{7\pi}{6}\right) \]The sine of \(-\frac{7\pi}{6}\) is the same as the sine of \(\pi - \frac{\pi}{6}\), which is \(-1/2\). Thus,\[ y = 11 \times \left(-\frac{1}{2}\right) = -\frac{11}{2} \]

Step 5: Conclusion

Combine the \(x\) and \(y\) values calculated to write the rectangular coordinates: \[ \left( -\frac{11\sqrt{3}}{2}, -\frac{11}{2} \right) \]

Key Concepts

Understanding Polar Coordinates

Polar coordinates consist of a pair

• - The first element, \( r \), indicates the radial distance from the origin.
• - The second element, \( \theta \), represents the angle from the positive x-axis.

These coordinates are particularly useful in systems where the relationship to a central point is more natural or simpler than using a rectangular grid.
For example, imagine a radar detecting airplanes; the position of each plane is often described more conveniently by how far away it is (\( r \)) and its direction (\( \theta \)).
In our given point, \( (11, -\frac{7\pi}{6}) \), the distance \( r \) is 11 units, and the angle \( \theta \) is \(-\frac{7\pi}{6}\), pointing in a standard direction. Converting these into rectangular coordinates requires some calculation.

Understanding Rectangular Coordinates

Rectangular, or Cartesian, coordinates describe positions on a plane using two values:

• The \( x \)-coordinate indicates the horizontal position.
• The \( y \)-coordinate indicates the vertical position.

Think of this system like graph paper, where you can locate any point by moving across (\( x \)) and then up or down (\( y \)).
Our task involves translating polar coordinates \((11, -\frac{7\pi}{6})\) into this system. We will compute new \( x \) and \( y \) values that directly describe the location in rectangular terms. This transformation helps applications like computer graphics to manipulate visual elements on the screen.

Using Trigonometric Functions for Conversion

Trigonometric functions, namely sine and cosine, are essential for converting between polar and rectangular coordinates.

• The formula \( x = r \cos(\theta) \) calculates the \( x \)-coordinate.
• The formula \( y = r \sin(\theta) \) calculates the \( y \)-coordinate.

In our example, we first calculate \( x \) using \( x = 11 \cos(-\frac{7\pi}{6}) \). Knowing trigonometric identities, \( \cos(-\theta) = \cos(\theta) \); hence, the calculations simplify, showing us \( \cos(-\frac{7\pi}{6}) = \cos(\pi - \frac{\pi}{6}) = -\sqrt{3}/2 \). For \( y \), we find \( y = 11 \sin(-\frac{7\pi}{6}) \). Here, \( \sin(-\theta) = -\sin(\theta) \), so \( \sin(-\frac{7\pi}{6}) = -1/2 \). These results lead us to our rectangular coordinates.

Angle Conversion and Its Role

Angles in polar coordinates often need conversion for trigonometric calculations. When working with angles such as \(-\frac{7\pi}{6}\), understanding their equivalent positives makes trigonometric operations straightforward.
Angles in radians where negatives might seem confusing, can often be converted by adding \(2\pi\) (a full circle's rotation) until the angle is positive.
In context, \(-\frac{7\pi}{6}\) becomes \(5\pi/6\) when traversed in the positive direction.
This conversion aids in better comprehension of directions and simplifies the application of functions in both mathematical and real-world scenarios. By mastering these conversions, you become proficient at predicting and interpreting motions and rotations effectively.