Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-8: Change the polar coordinates to rectangular coordinates. $$ \left(6, \arctan \frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{24}{5}, \frac{18}{5}\right)\).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form (\(r\), \(\theta\)), where \(r\) is the radius or distance from origin, and \(\theta\) is the angle in radians from the positive x-axis.

Step 2: Identify the Given Information

Here, \(r = 6\) and \(\theta = \arctan\left(\frac{3}{4}\right)\).

Step 3: Relationship to Rectangular Coordinates

To convert from polar to rectangular coordinates, use the equations: \(x = r \cos \theta\)and \(y = r \sin \theta\).

Step 4: Determine \(\theta\) Using Tangent

Since \(\theta = \arctan\left(\frac{3}{4}\right)\), we know that \(\tan \theta = \frac{3}{4}\). We need to find \(\cos \theta\) and \(\sin \theta\) that satisfy this ratio.

Step 5: Use Pythagorean Theorem

Create a right triangle where the opposite side is 3 and the adjacent side is 4. Since \(\tan \theta = \frac{3}{4}\), the hypotenuse \(h\) can be found using:\[h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\].Thus, \(\cos \theta = \frac{4}{5}, \sin \theta = \frac{3}{5}\).

Step 6: Calculate \(x\) and \(y\)

Using the relationships:\(x = r \cos \theta = 6 \times \frac{4}{5} = \frac{24}{5}\)and \(y = r \sin \theta = 6 \times \frac{3}{5} = \frac{18}{5}\).

Step 7: Final Rectangular Coordinates

Thus, the rectangular coordinates are:\(x = \frac{24}{5}\)and \(y = \frac{18}{5}\).Hence, the rectangular coordinates are \(\left(\frac{24}{5}, \frac{18}{5}\right)\).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative way of representing points in a plane using a radial distance and an angle. Think of it as describing a point by saying how far it is and in which direction you need to go. This system is especially useful in scenarios known for their circular symmetry, such as in physics dealing with orbits.

In polar coordinates, a point is denoted as \((r, \theta)\). Here, \(r\) is the distance from the origin, and \(\theta\) is the angle measured from the positive x-axis in the counter-clockwise direction.

Key points to remember:

• \(r\) can be any non-negative number, representing the radius.
• \(\theta\) is usually expressed in radians but can also be in degrees.
• The origin itself can have a coordinate of \((0, \theta)\), where \(\theta\) can be any angle because radius is zero.

Converting from polar coordinates to another form like rectangular can help solve different types of problems more efficiently.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using the familiar \((x, y)\) format. This system is named after the philosopher and mathematician René Descartes.

In this system: - The \(x\)-coordinate specifies horizontal position; it can be positive, negative, or zero.- The \(y\)-coordinate specifies vertical position; similarly, it can be positive, negative, or zero.

These coordinates are used mostly in straightforward, everyday geometry and algebra tasks:

• It's easy to plot points and graph equations.
• They allow simple derivation of straight-line equations using y-intercepts and slopes.
• They are the default when dealing with linear relationships in the plane.

To move between polar and rectangular coordinates relies on leveraging trigonometric relationships.

Trigonometric Functions

Trigonometric functions are fundamental in linking angular and linear dimensions. They enable the transition between polar and rectangular coordinates.

Important functions include:

• Sine (\(\sin\)): It relates an angle in a right triangle to the ratio of the opposite side to the hypotenuse. \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
• Cosine (\(\cos\)): It defines the ratio of the adjacent side to the hypotenuse. \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
• Tangent (\(\tan\)): Relates an angle to the ratio of the opposite side to the adjacent side. \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).

In coordinate conversion, you directly use cosine and sine to find the \(x\) and \(y\) values from \(r\) and \(\theta\), using:

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

Such formulas make trigonometry a cornerstone for navigating between different coordinate systems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry relating the sides of a right triangle. According to this theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides: \[ a^2 + b^2 = c^2 \]In the context of converting coordinates, you use this theorem to discover relationships between trigonometric ratios.

For instance, in our exercise:

• Given \(\tan \theta = \frac{3}{4}\), create sides with lengths 3 and 4.
• Determine the hypotenuse as 5 by applying the theorem: \(\sqrt{3^2 + 4^2} = \sqrt{25} = 5\).

This allows calculation of sine and cosine:

• \(\cos \theta = \frac{4}{5}\)
• \(\sin \theta = \frac{3}{5}\)

These values are critical in transforming polar coordinates like \((6, \arctan(\frac{3}{4}))\) into rectangular coordinates. Understanding the Pythagorean Theorem in this context helps you easily manage coordinate conversion tasks.