Problem 28 - Algebra and Trigonometry

Question

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=-\frac{\pi}{2} $$

Step-by-Step Solution

Answer

The terminal point is \((0, -1)\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. Any angle, such as the given angle \(t = -\frac{\pi}{2}\), will correspond to a point \(P(x, y)\) on this circle where \(x^2 + y^2 = 1\).

Step 2: Determine the Angle on the Unit Circle

The angle \(t = -\frac{\pi}{2}\) signifies a rotation in the clockwise direction because it is negative. Starting from the positive x-axis, rotating \(-\frac{\pi}{2}\) radians places us on the negative y-axis.

Step 3: Identify the Coordinates on the Unit Circle

When the angle lands on one of the axes, the coordinates are easier to infer. At \(t = -\frac{\pi}{2}\), the point on the unit circle is directly down on the negative y-axis. Thus, the terminal point \(P(x, y)\) is \((0, -1)\).

Key Concepts

Trigonometric Angles

Trigonometric angles are essential in understanding the relationship between angles and the points on a circle, especially on the unit circle. These angles are usually measured in radians, which is a standard unit in trigonometry and originates from the circle's radius. For example, a full circle is equal to \(2\pi\) radians, and a right angle is \(\frac{\pi}{2}\) radians.

Angles can be positive or negative. A positive angle signifies a counter-clockwise rotation from the positive x-axis, while a negative angle represents a clockwise rotation. In the case of \(t = -\frac{\pi}{2}\), this means the angle rotates clockwise by half a right angle. Understanding this concept helps in precisely locating terminal points on the unit circle.

Understanding trigonometric angles helps us in identifying precise positions on the circle, allowing for accurate mathematical calculations and representations.

Terminal Point

A terminal point is the endpoint of the radius that pivots on the origin in the unit circle when rotated by a specific angle. The terminal point is vital because it has coordinates \(P(x, y)\), which lie on the unit circle. These coordinates help determine specific values of trigonometric functions, like sine and cosine, for the corresponding angle.

To find the terminal point for \(t = -\frac{\pi}{2}\), you imagine rotating the radius from the positive x-axis by \(-\frac{\pi}{2}\) radians. This angle leads the radius to point directly downwards along the negative y-axis. Thus, this specific rotation brings the terminal point to \((0, -1)\), where the x-coordinate is 0, and the y-coordinate is -1.

The concept of terminal points allows us to easily identify trigonometric values and their relationships according to the angle given.

Coordinate Plane

The coordinate plane is a two-dimensional space made up of an x-axis (horizontal) and a y-axis (vertical) that intersect at a point called the origin. The use of the coordinate plane is significant in many areas of mathematics, including geometry and trigonometry.

When considering the unit circle on the coordinate plane, its center at the origin \((0,0)\) and radius of 1 help define every possible point using coordinates \((x, y)\). Each point on the unit circle corresponds to a unique angle made with the positive x-axis, demonstrating the relationship between geometric and algebraic concepts. In the exercise,\(t = -\frac{\pi}{2}\) indicates a specific point, \((0, -1)\), on this coordinate plane.

The coordinate plane thus provides a system through which we can visualize and solve trigonometric problems by using coordinates to represent specific angles and points.