Problem 4 - 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 \(\theta .\) \(O A=2, \theta=30^{\circ}\)

Step-by-Step Solution

Answer

The coordinates of point A are \((\sqrt{3}, 1)\).

Step 1: Convert Degree to Radians

To work with angles in trigonometry, usually, we use radians instead of degrees. First, convert the angle from degrees to radians using the formula: \[\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\]For \(\theta = 30^{\circ}\):\[30 \times \frac{\pi}{180} = \frac{\pi}{6}\]

Step 2: Identify the Reference Triangle

For an angle \(\theta = 30^{\circ}\) or \(\theta = \frac{\pi}{6}\), we recognize the 30-60-90 reference triangle. The hypotenuse is the radius \(OA = 2\), and we need to find the coordinates using the properties of this triangle.

Step 3: Calculate the Coordinate Values

In a 30-60-90 triangle, the ratio of the sides opposite the 30-degree angle, 60-degree angle, and the hypotenuse are 1: \(\sqrt{3}\) : 2. For \(OA = 2\):- The x-coordinate (adjacent to the 30-degree angle) = \(2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}\).- The y-coordinate (opposite the 30-degree angle) = \(2 \cdot \frac{1}{2} = 1\).

Step 4: Express Coordinates in Simplest Radical Form

Using the calculations from Step 3, express the coordinates of the point \(A\) in simplest radical form. The coordinates of point \(A\) are:\(A(\sqrt{3}, 1)\).

Key Concepts

Understanding Radian Conversion

When dealing with trigonometry, we often need to convert angles from degrees to radians. This is crucial as radians are the standard unit of angular measure used in many branches of mathematics. To convert degrees to radians, you use the formula:

• \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \)

For example, to convert \(30^{\circ}\) to radians, substitute 30 for \(\theta_{\text{degrees}}\) in the formula:\[ 30 \times \frac{\pi}{180} = \frac{\pi}{6} \]This \(rac{ extrm{pi}}{6}\) tells you that each slice of the circle formed by this angle in radians aligns perfectly with standardized trigonometric functions, like sine and cosine. Understanding how to convert between these two systems allows for greater flexibility in solving trigonometric problems.

Exploring the 30-60-90 Triangle

In trigonometry, the 30-60-90 triangle is a special type of right triangle with useful properties due to its specific angle measures. It consists of angles measuring 30 degrees, 60 degrees, and 90 degrees. This triangle is unique because it maintains a consistent ratio of side lengths:

• 1 (opposite the 30-degree angle)
• \(\sqrt{3}\) (opposite the 60-degree angle)
• 2 (the hypotenuse or longest side)

Knowing the ratios of these sides is incredibly helpful in solving trigonometric problems involving angles of \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). When given a hypotenuse, you can easily algebraically manipulate these ratios to find the lengths of the other sides. For a hypotenuse of 2, as found in the given exercise, you can scale the side ratios to find:

• The side opposite the 30-degree angle is \(1 \times 2/1 = 1\).
• The side opposite the 60-degree angle is \(\sqrt{3} \times 2/2 = \sqrt{3}\).

Using them helps derive precise coordinates on a unit circle, facilitating the solving of more complex trigonometric functions.

Coordinates in Simplest Radical Form

Finding coordinates in simplest radical form primarily involves calculating the position of a point using its relationship to the angle and the radius (hypotenuse in our triangle scenario). From previous steps, using a 30-60-90 triangle, we already know:

• The x-coordinate is \(\sqrt{3}\), which represents the length that is adjacent to the 30-degree angle.
• The y-coordinate is 1, illustrating the length opposite the 30-degree angle.

Since these sides derive directly from the radian angle designation and use defined triangular side ratios, expressing them in simplest radical form involves reflecting these pure geometric truths without any further modifications or simplifications. For instance, the calculation for the x-coordinate is:\[ 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}\]And for the y-coordinate:\[ 2 \cdot \frac{1}{2} = 1\]Thus, the coordinates of point \(A\) on the terminal side of a 30-degree angle, with radius 2, become \(A(\sqrt{3}, 1)\). Keeping the coordinates in simplest radical form is crucial, as it offers a clear depiction of the spatial relationships dictated by the geometry involved.