Problem 6 - 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 \(\theta .\) \(O A=8, \theta=120^{\circ}\)

Step-by-Step Solution

Answer

The coordinates of point A are \((-4, 4\sqrt{3})\).

Step 1: Understanding the Reference Triangle

An angle of 120° is in the second quadrant. The reference angle for 120° is 60° since it is the difference between 180° and 120°.

Step 2: Finding Coordinates using the Unit Circle

In the unit circle, the coordinates corresponding to a 60° angle are \((\frac{1}{2}, \frac{\sqrt{3}}{2})\). Because the angle 120° lies in the second quadrant, the x-coordinate is negative, resulting in \((-\frac{1}{2}, \frac{\sqrt{3}}{2})\).

Step 3: Scaling to the Correct Radius

The radius is given as 8, so we need to scale the unit circle coordinates accordingly. Multiply both coordinates by 8: \((-\frac{1}{2} \times 8, \frac{\sqrt{3}}{2} \times 8)\).

Step 4: Simplifying the Coordinates

Calculate the products: \(-\frac{1}{2} \times 8 = -4\) and \(\frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3}\). The final coordinates are \((-4, 4\sqrt{3})\).

Key Concepts

Reference Angle

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always positive and lies between 0° and 90°, making it easier for us to understand the position of the angle within the unit circle. For example, if you have an angle of 120°, the reference angle helps to provide a simpler context for calculations. This is because the reference angle for 120° is 60°, found by subtracting from 180° (since 120° lies in the second quadrant).

• It helps to visualize the angle’s position relative to standard x and y coordinates.
• A crucial tool for calculations involving trigonometric functions.
• Always positive, making them easy to work with.

Understanding reference angles is vital for navigating the complexities of trigonometry smoothly!

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental concept in trigonometry, helping to visualize and calculate trigonometric functions like sine, cosine, and tangent. Each point on the unit circle has coordinates that represent the cosine and sine of the angle formed from the positive x-axis.

• Aids in finding function values for various angles.
• Helps in calculating coordinates by using scale multiplication for different radii.
• Provides reference values for different angles, simplifying trigonometric equations.

For the angle 60°, the coordinates on a unit circle are (\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\)). This understanding becomes particularly useful when needing to adjust these values for other radii, as seen in our example where the radius was scaled to 8.

Coordinates

Coordinates are crucial when navigating within the coordinate system, indicating the exact position of a point on this grid. With angles, particularly those on the terminal side, their placement within a coordinate system is defined by x and y coordinates.Here’s how it applies to trigonometry:

• For 120°, the coordinates derived from the unit circle and then adjusted for the radius are (-\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\)).
• To find the actual coordinates, multiply these unit circle values by the radius, here 8.
• This yields final coordinates for the problem as (-4, 4\(\sqrt{3}\)).

So, when trigonometry refers to the coordinates, it often involves a series of transformations starting with unit circle values followed by any necessary scaling.

Terminal Side of an Angle

The terminal side of an angle is a key concept in trigonometry. It represents the final position of a ray after it has rotated around its initial position. Understanding where this lies helps in determining the angle's placement and subsequently, its coordinates. Here’s what finding the terminal side involves:

• Begins with placing the angle in standard position, starting from the positive x-axis.
• The endpoint of this angle lies on the terminal side after a specific degree of rotation.
• This aspect helps determine the correct sign of unit circle coordinates, critical in different quadrants.

For our angle of 120°, the terminal side lands in the second quadrant. Therefore, the x-coordinate is negative, while the y-coordinate remains positive, leading to correct calculations for positions and trigonometric function values.