Problem 33 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(-1, \pi+\arctan \left(\frac{3}{4}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{4}{5}, -\frac{3}{5}\right)\).

Step 1: Understand Polar Coordinates

Polar coordinates are given as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

Step 2: Assign Variables

For the point \((-1, \pi + \arctan\left(\frac{3}{4}\right))\), \(r = -1\) and \(\theta = \pi + \arctan\left(\frac{3}{4}\right)\).

Step 3: Calculate the Angle in Rectangular Form

\( \theta = \pi + \arctan\left(\frac{3}{4}\right) \) results in an angle in the third quadrant since adding \(\pi\) to the angle moves it 180 degrees around the circle.

Step 4: Convert to Rectangular Coordinates

Use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, \(x\) and \(y\) represent the rectangular coordinates.

Step 5: Substitute and Simplify for x

\(x = -1 \times \cos(\pi + \arctan\left(\frac{3}{4}\right)) = -\cos\left( \arctan\left(\frac{3}{4}\right) + \pi\right)\). Using the identity \(\cos(a + b) = -\cos(a)\) when \(b = \pi\), we get \(x = \cos\left( \arctan\left(\frac{3}{4}\right)\right) = \frac{4}{5}\).

Step 6: Substitute and Simplify for y

\(y = -1 \times \sin(\pi + \arctan\left(\frac{3}{4}\right)) = -\sin\left( \arctan\left(\frac{3}{4}\right) + \pi\right)\). Using the identity \(\sin(a + b) = -\sin(a)\) when \(b = \pi\), we get \(y = - \sin\left( \arctan\left(\frac{3}{4}\right)\right) = -\frac{3}{5}\).

Step 7: Write the Rectangular Coordinates

The rectangular coordinates are \((x, y) = \left(\frac{4}{5}, -\frac{3}{5}\right)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical relationships that simplify expressions involving trigonometric functions like sine and cosine. They are crucial for converting polar coordinates to rectangular coordinates. Two fundamental identities are used in this process:

• The cosine of an angle added to \( \pi \) is the negative of the cosine of the original angle: \( \cos(a + \pi) = -\cos(a) \).
• The sine of an angle added to \( \pi \) is the negative of the sine of the original angle: \( \sin(a + \pi) = -\sin(a) \).

These identities reflect the symmetry of the trigonometric circle and help when angles go beyond the initial 0 to \( 2\pi \) range. Since polar coordinates involve angles, these identities allow for straightforward conversion to rectangular coordinates by adjusting the trigonometric values used in calculations. By knowing these identities, you can confidently work through the conversion process.

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Unlike rectangular coordinates, which specify a point based on horizontal and vertical distances from an origin, polar coordinates define a point with:

• \( r \): the radial distance from the pole or origin.
• \( \theta \): the angle measured from the positive x-axis.

In this exercise, the polar coordinates for the point are \((-1, \pi + \arctan\left(\frac{3}{4}\right))\). Here, \( r = -1 \) means the point is one unit away from the origin but in the opposite direction, making it essential to consider the angle \( \theta = \pi + \arctan\left(\frac{3}{4}\right) \), which places the point in the third quadrant. Understanding how these components work together is crucial for effectively converting to rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point by its horizontal and vertical position in a plane, relative to an origin. These are given as \((x, y)\), where:

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

In the conversion of the polar point \((-1, \pi + \arctan\left(\frac{3}{4}\right))\) to rectangular coordinates, we use the identities to simplify the process:

• \( x = r \cos(\theta) = -1 \times \cos(\pi + \arctan\left(\frac{3}{4}\right)) = \frac{4}{5} \)
• \( y = r \sin(\theta) = -1 \times \sin(\pi + \arctan\left(\frac{3}{4}\right)) = -\frac{3}{5} \)

By substituting and calculating these values, using the trigonometric identities mentioned earlier, we achieve the rectangular coordinates \((\frac{4}{5}, -\frac{3}{5})\). This conversion process highlights the practical application of understanding both coordinate systems and how they relate to each other.