Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-22\) , for each given angle in standard position, determine to the nearest tenth the coordinates of the point where the terminal side intersects the unit circle. $$ -180^{\circ} $$

Step-by-Step Solution

Answer

The coordinates are \((-1, 0)\).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) on a coordinate plane. Every point on the circle satisfies the equation \(x^2 + y^2 = 1\). Angles in standard position are measured from the positive x-axis.

Step 2: Determine the Reference Angle

For the given angle \(-180^{\circ}\), start by recognizing that negative angles are measured clockwise from the positive x-axis. Since \(180^{\circ}\) translates to a half-circle, rotating \(-180^{\circ}\) results in the terminal side landing on the negative x-axis.

Step 3: Identify the Coordinates

The terminal side at \(-180^{\circ}\) is a straight line along the negative x-axis. On the unit circle, this corresponds to the point \((-1, 0)\) because the radius (1 unit in length) defines the circumference of the circle.

Step 4: Round the Coordinates

The coordinates \((-1, 0)\) are already at their simplest form and don't require further rounding. If they needed to be rounded to the nearest tenth, they would still remain \((-1.0, 0.0)\).

Key Concepts

Standard Position

In trigonometry, angles in the standard position are crucial for understanding how they relate to the unit circle. When we talk about the standard position, we're referring to angles whose vertex is at the origin of a coordinate plane. The initial side of the angle lies along the positive x-axis. This positioning is important because it provides a consistent way to reference angles, making it easier to determine their sine and cosine values on the unit circle.

When an angle is in the standard position, we observe how the terminal side rotates from the positive x-axis. The rotation can be either counterclockwise, which denotes positive angles, or clockwise, marking negative angles. This consistent framework allows mathematicians to easily describe and analyze the positions and movements of angles related to the circle.

Negative Angles

Negative angles, in trigonometry, are measured by rotating clockwise from the positive x-axis. Unlike positive angles which rotate counterclockwise, a negative angle signifies a reversal of this direction. Understanding this concept helps simplify calculations when dealing with various mathematical and real-world applications.

To illustrate, a negative angle like \(-180^{\circ}\) involves moving in a clockwise direction, halfway around the circle before stopping at the negative x-axis. This characteristic helps determine specific points on the unit circle, which are instrumental in problems involving trigonometric functions and their applications.

Reference Angle

A reference angle is the positive acute angle that one forms with the x-axis. It's essential for simplifying trigonometric calculations, as it provides a basis for analyzing angles in any quadrant. Think of it as a helper angle that always reveals how far an angle strays from the closest x-axis.

Even when dealing with negative angles, the reference angle remains relevant. Take \(-180^{\circ}\) as an example. While the rotation lands you on the negative x-axis, the reference angle remains \(0^{\circ}\) since you're exactly on the axis itself. For angles not perfectly aligned with the x-axis, calculating the reference angle involves adjusting to always result in a positive measurement, aiding in easier computation and understanding.

Coordinate Plane

The coordinate plane is a foundational tool in mathematics, providing a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). When related to the unit circle, it's especially useful for pinpointing the exact location where the terminal side of an angle intersects with the circle.

In our context, the unit circle is centered at the origin \((0,0)\) and has a radius of 1. This means any point on the circle satisfies the equation \(x^2 + y^2 = 1\). The coordinates of these points are essential in determining angles' exact positions and their corresponding sine and cosine values.

• For \(-180^{\circ}\), this places us directly on the negative x-axis, intersecting the circle precisely at \((-1,0)\).
• Understanding this intersection helps link geometric visualization with algebraic equations in a cohesive manner.