Problem 5 - 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 \(\theta .\) \(O A=6, \theta=90^{\circ}\)

Step-by-Step Solution

Answer

The coordinates of point \( A \) are \( (0, 6) \).

Step 1: Understanding the Problem

Given is a point \( A \) on the terminal side of an angle \( \theta \) in standard position with \( OA = 6 \) and \( \theta = 90^\circ \). We need to find the coordinates of point \( A \).

Step 2: Understanding Angle in Standard Position

An angle is in standard position when its vertex is at the origin \((0,0)\), and the initial side is along the positive \(x\)-axis. For \( \theta = 90^\circ \), the terminal side is along the positive \(y\)-axis.

Step 3: Determine the Coordinates Using the Radius

The distance from the origin \((0,0)\) to point \( A \) is 6 units (\( OA=6 \)). Since \( \theta = 90^\circ \) directs the radius upwards along the \(y\)-axis, \( A \) has coordinates \((0, 6)\), because the entire length is vertical along the \(y\)-axis.

Key Concepts

Angles in Standard Position

An angle is defined to be in its standard position when its vertex is located at the origin of the coordinate system, specifically at yet at **coordinates**: - (0,0), - and with its initial side resting along the positive x-axis. When we talk about angles in standard position, the key idea is to think of the angle as starting from this positive x-axis and rotating counterclockwise, unless specified otherwise. This is the orientation we use for measuring angles in geometry and trigonometry. For example, if an angle measures 90° in standard position, the rotation from the x-axis will land the terminal side of the angle on the positive y-axis. This aligns perfectly with the typical top view of the coordinate plane. Always remember:

• The vertex is at the origin.
• The initial side lies on the positive x-axis.
• Angles are measured from the initial side to the terminal side, often counterclockwise.

Coordinates of Points

Coordinates are a way of determining a specific location in a 2-dimensional space, often described in terms of - distance - direction from a reference point, which is the origin in the Cartesian coordinate system. To express the coordinates of a point, we use an ordered pair - (x, y) where - **x** is the horizontal distance from the origin - **y** is the vertical distance from the origin. For example, in a problem where an angle is in standard position and opens up by 90 degrees, the terminal side lays flat on the y-axis. Let's consider our Point A for the angle given: - The horizontal component (x) is 0 because it doesn't deviate horizontally from the y-axis. - The vertical component (y) is the length of the radius, which is 6 units. This gives point A the coordinate (0, 6). Remember:

• The first number corresponds to the x-coordinate (horizontal)
• The second number corresponds to the y-coordinate (vertical)
• When an angle opens up perfectly vertically or horizontally, one of the coordinates becomes the radius while the other is 0.

Right Angle

A right angle is one of the most common and recognizable angles in geometry. It measures exactly 90 degrees, and can also be described in radians as \(\frac{\pi}{2}\).When envisioning a right angle in the coordinate system, imagine starting at the positive x-axis and rotating a quarter turn to land on the positive y-axis. This forms an "L" shape, featuring: - **Two intersecting lines** that form a square corner.A right angle is often used as a reference angle in trigonometry because it helps easily measure and calculate various elements like sine and cosine values. This is where 90-degree angles shine as they facilitate our understanding of how every other angle spatially relates.In standard math problems, like finding the coordinates of a point along a terminal side or circle:

• A right angle signifies that the distance will extend vertically or horizontally from the origin.
• All points along the vertical or horizontal line form with the origin will share the same y or x value, respectively.