Problem 9 - Algebra 2 and Trigonometry

Question

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=9, \theta=150^{\circ}\)

Step-by-Step Solution

Answer

The coordinates of point A are \((-\frac{9\sqrt{3}}{2}, \frac{9}{2})\).

Step 1: Understand the Terminal Side Point

The point on the terminal side of an angle in standard position is given in polar coordinates as \((r, \theta)\), where \(r\) is the distance from the origin (radius or hypotenuse) and \(\theta\) is the angle. We need to convert these into Cartesian coordinates \((x, y)\). In this exercise, we are given \(O A = 9\) (equivalent to \(r = 9\)) and \(\theta = 150^{\circ}\).

Step 2: Convert to Cartesian Coordinates

Convert polar coordinates \((9, 150^{\circ})\) into Cartesian coordinates \((x, y)\). Using the formulas: - \(x = r \cdot \cos(\theta)\) - \(y = r \cdot \sin(\theta)\) Start by evaluating the trigonometric functions. We know that \(\cos(150^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2}\) and \(\sin(150^{\circ}) = \sin(30^{\circ}) = \frac{1}{2}\).

Step 3: Calculate x-coordinate

Calculate \(x\) using the formula: \(x = 9 \cdot \cos(150^{\circ}) = 9 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{9\sqrt{3}}{2}\).

Step 4: Calculate y-coordinate

Calculate \(y\) using the formula: \(y = 9 \cdot \sin(150^{\circ}) = 9 \cdot \frac{1}{2} = \frac{9}{2}\).

Step 5: Express in Simplest Radical Form

With the calculations complete, the Cartesian coordinates of point \(A\) are \(\left(-\frac{9\sqrt{3}}{2}, \frac{9}{2}\right)\). These coordinates are already in their simplest radical form.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics to relate the angles and sides of triangles. They help in converting polar coordinates to Cartesian coordinates. Specifically, when we deal with an angle \( \theta \), we often use the basic trigonometric functions such as sine and cosine.To convert polar coordinates \( (r, \theta) \) into Cartesian coordinates \( (x, y) \), we use:

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

In the given problem, the angle \( \theta = 150^{\circ} \) relates to the trigonometric functions as follows:

• \( \cos(150^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2} \)
• \( \sin(150^{\circ}) = \sin(30^{\circ}) = \frac{1}{2} \)

These functions help us pinpoint the exact horizontal and vertical distances from the origin.

Simplest Radical Form

When converting to or expressing anything in simplest radical form, the aim is to simplify expressions but still keep the radicals. For coordinates, especially when derived from trigonometric calculations, using simplest radical form helps maintain precision.In the problem, we have the trigonometric results:

• \( x = 9 \cdot \cos(150^{\circ}) = 9 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{9\sqrt{3}}{2} \)
• \( y = 9 \cdot \sin(150^{\circ}) = 9 \cdot \frac{1}{2} = \frac{9}{2} \)

Each component, \( x \) and \( y \), is expressed in simplest radical form. Simplifying into radicals keeps expressions exact versus approximating with decimals, which could lead to inaccuracies.

Coordinates in Standard Position

Coordinates in standard position refer to points placed in the coordinate plane based on angles measured from the positive x-axis. When an angle is drawn in standard position:

• Its vertex is at the origin \( (0,0) \).
• Its initial side lies along the positive x-axis.

The terminal side of the angle gives the direction and distance to a point. If you know the angle and radius, you can easily determine this point's location using polar coordinates \( (r,\theta) \). By using the trigonometric functions explained earlier, you can convert these into Cartesian coordinates to pinpoint exact positions in a rectangular grid. In this example, knowing \( O A = 9 \) and \( \theta = 150^{\circ} \) enables the conversion to the Cartesian coordinates \( \left(-\frac{9\sqrt{3}}{2}, \frac{9}{2}\right) \). This showcases the relationship between angle measurements, directional placement, and coordinate conversion.