Problem 10 - 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 \(\theta .\) \(O A=25, \theta=210^{\circ}\)

Step-by-Step Solution

Answer

The coordinates of point \( A \) are \( \left( -\frac{25\sqrt{3}}{2}, -\frac{25}{2} \right) \).

Step 1: Understand the Problem

The problem involves an angle \( \theta = 210^{\circ} \) in standard position, and a point \( A \) on the terminal side such that the distance from the origin \( O \) to \( A \) (\( OA \)) is 25. We need to find the coordinates of point \( A \).

Step 2: Find Reference Angle

The reference angle for \( 210^{\circ} \) is found by subtracting \( 180^{\circ} \) from \( 210^{\circ} \), which gives \( 30^{\circ} \). This helps us identify the coordinates using known values for a 30-degree reference angle.

Step 3: Determine Coordinates using Trigonometric Functions

The terminal side at \( 210^{\circ} \) lies in the third quadrant, where both x- and y-coordinates are negative. Using trigonometric functions: - \( x = r \cdot \cos(\theta) = 25 \cdot \cos(210^{\circ}) = 25 \cdot (-\frac{\sqrt{3}}{2}) = -\frac{25\sqrt{3}}{2} \) - \( y = r \cdot \sin(\theta) = 25 \cdot \sin(210^{\circ}) = 25 \cdot (-\frac{1}{2}) = -\frac{25}{2} \)

Step 4: State the Coordinates in Simplest Radical Form

The coordinates of point \( A \) are found using the calculations from Step 3:\( A\left( -\frac{25\sqrt{3}}{2}, -\frac{25}{2} \right) \). This is already in simplest radical form.

Key Concepts

trigonometric functions

Trigonometric functions are tools we use to explore relationships in triangles, especially right-angled ones. When we place these functions in a coordinate system, they help us find the coordinates of a point on a circle's circumference, based on an angle's measurement. Here, we're dealing with an angle, \( \theta = 210^{\circ} \), which is in the standard position.

• The cosine function, often written as \( \cos(\theta) \), gives the x-coordinate of a point on a circle of radius 1.
• The sine function, \( \sin(\theta) \), offers the y-coordinate under the same circumstances.

By utilizing these functions, we can determine the exact "footprint" of point \( A \) on this circle. For our specific problem, knowing that \( r = 25 \), we multiply the standard sine and cosine values of the angle to scale up to the actual circle we are looking at.

standard position

An angle is said to be in "standard position" when its vertex is at the origin of the coordinate system, and its initial side lies along the positive x-axis. This is a foundational concept in trigonometry, as it helps standardize how angles are measured and visualized.When you have an angle like \( \theta = 210^{\circ} \), you start measuring from the positive x-axis in a counterclockwise direction. As the angle increases, the terminal side will sweep across different quadrants of the circle. A full circle is 360 degrees. Therefore, \( 210^{\circ} \) places the terminal side in the third quadrant.Understanding the standard position is essential, especially when you are trying to solve problems involving angles and circles. This systematic way of measuring provides a consistent framework to analyze and solve trigonometric problems.

terminal side

The term "terminal side" refers to the final position of the angle's ray after the angle has been drawn from its initial side. This is crucial because it determines which quadrant the angle resides in, thereby affecting the signs (+ or -) of the trigonometric functions involved.In the case of \( \theta = 210^{\circ} \), the terminal side is located in the third quadrant of the coordinate plane. For angles in the third quadrant:

• Both x and y coordinates are negative.
• The reference angle can guide us to find these coordinates more easily by working with known sine and cosine values.

Using the reference angle of 30 degrees from \( 210^{\circ} \), we are able to determine the points on the circumference of a circle, scaled by the radius of 25. This ensures the coordinates we find reflect the position and orientation of the angle's terminal side accurately.