Problem 20 - Algebra and Trigonometry

Question

15-20 \(\mathbf{m}\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(x\) -coordinate of \(P\) is \(-\frac{2}{5},\) and \(P\) lies above the \(x\) -axis.

Step-by-Step Solution

Answer

Point P is \\left(-\frac{2}{5}, \frac{\sqrt{21}}{5}\right)\\.

Step 1: Identify the properties of the unit circle

A unit circle is a circle with a radius of 1, centered at the origin \(0,0\). Therefore, any point \(P(x,y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\) since the radius is always 1.

Step 2: Substitute the given x-coordinate

We are given that the \(x\)-coordinate of point \(P\) is \(-\frac{2}{5}\). Substitute \(x = -\frac{2}{5}\) into the unit circle equation: \((-\frac{2}{5})^2 + y^2 = 1\).

Step 3: Simplify and solve for y²

Square the \(x\)-coordinate: \((-\frac{2}{5})^2 = \frac{4}{25}\). Substitute this into the equation to get \(\frac{4}{25} + y^2 = 1\). Subtract \(\frac{4}{25}\) from 1 to solve for \(y^2\): \(y^2 = 1 - \frac{4}{25} = \frac{21}{25}\).

Step 4: Determine the value of y

Since \(P\) is above the \(x\)-axis, \(y\) is positive. Therefore, \(y = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}\).

Step 5: Write the coordinates of point P

Combine the values of \(x\) and \(y\) to express the coordinates of point \(P\): \((-\frac{2}{5}, \frac{\sqrt{21}}{5})\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a vital part of mathematics that enables us to study geometrical shapes using algebra. In essence, it connects algebra and geometry using a coordinate system. In two dimensions, we use an x-axis and a y-axis, which helps in describing positions.

• The origin is the point where the axes intersect, located at \( (0, 0) \).
• Each point on the plane is represented by a pair of numbers called coordinates.
• The x-coordinate measures the horizontal position, while the y-coordinate measures the vertical position.

Understanding Points: Points like \((x, y)\) on a coordinate plane show us exact locations. They help us understand where a point lies relative to the origin.
The Role of Coordinates: Each coordinate corresponds to a specific distance from the axles, making it straightforward to visualize points, shapes, and distances. This is especially useful in problems involving circles and other geometric figures where precise measurements are needed.

Circle Equations

Circle equations describe all the points that form a circle. For a circle at the origin with a radius of 1, known as the unit circle, the equation is \( x^2 + y^2 = 1 \). This equation encapsulates all the points (x, y) that are exactly one unit away from the origin.

• The general formula for any circle centered at the origin is \( x^2 + y^2 = r^2 \), where \( r \) is the radius.
• If a circle isn’t centered at the origin, its equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle.

Unit Circle Specifics: In our example, the unit circle facilitates easy calculation since its radius is 1. Therefore, finding coordinates simply involves solving the equation \( x^2 + y^2 = 1 \) for either x or y using the values we have. Each point on this circle has this unique property that simplifies many trigonometric calculations, making it a powerful tool in both geometry and trigonometry.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry related to the geometry of the unit circle. It states that for a point on the unit circle with an angle \( \theta \), the relationship \( \sin^2(\theta) + \cos^2(\theta) = 1 \) always holds true. This identity is derived from the Pythagorean theorem, adapted to fit within the context of a circle.

• The formula mirrors the circle equation: \( x^2 + y^2 = 1 \), where \( x = \cos(\theta) \) and \( y = \sin(\theta) \).
• This shows that for any angle, the sum of the squares of its cosine and sine is 1.

Application in Circle Geometry:When working on problems involving unit circles, the Pythagorean Identity allows us to easily find unknown values. If one coordinate of a point on the unit circle is known, the other can be determined using this identity.
This principle was evident in our exercise, where we derived the y-coordinate using the known x-coordinate. Additionally, since our point lies above the x-axis, the sine (y-value) remains positive, as expected from the unit circle's properties.