Problem 18 - Algebra and Trigonometry

Question

\(13-18=\) 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 \(P\) has coordinates \(\left(-\frac{2}{5}, \frac{\sqrt{21}}{5}\right)\).

Step 1: Understanding the Unit Circle

The equation of the unit circle is given by \(x^2 + y^2 = 1\). This will help us find the value of \(y\) if \(x\) is known.

Step 2: Substitute the Given x-coordinate

We know the x-coordinate of point \(P\) is \(-\frac{2}{5}\). Substitute \(x = -\frac{2}{5}\) into the equation of the unit circle: \[\left(-\frac{2}{5}\right)^2 + y^2 = 1\].

Step 3: Solve for y²

Calculate \(\left(-\frac{2}{5}\right)^2 = \frac{4}{25}\). Substitute this back into the circle equation: \[\frac{4}{25} + y^2 = 1\].

Step 4: Isolate y²

Subtract \(\frac{4}{25}\) from both sides of the equation: \[y^2 = 1 - \frac{4}{25}\].

Step 5: Simplify the Equation

Calculate the right-hand side: \[y^2 = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}\].

Step 6: Solve for y

We have \(y^2 = \frac{21}{25}\). Since \(P\) lies above the x-axis, \(y > 0\). Thus, \(y = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}\).

Step 7: Conclusion on Coordinates

The coordinates of point \(P\) are \(\left(-\frac{2}{5}, \frac{\sqrt{21}}{5}\right)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is an important branch of mathematics that involves the use of algebraic equations to describe geometric shapes and properties. The unit circle, a key concept in coordinate geometry, is a circle with a radius of one unit, centered at the origin of a Cartesian coordinate system.

• The unit circle is represented by the equation: \[ x^2 + y^2 = 1 \]
• Any point \(P(x, y)\) on the unit circle satisfies this equation.
• The circle is symmetric about the x-axis and the y-axis, making it a useful tool for solving problems involving symmetry and periodicity in trigonometry.

Understanding coordinate geometry allows students to solve various problems related to circles and other geometric figures effectively.

Equation of Circle

An equation of a circle helps in defining all the points that lie on the perimeter of the circle. For the unit circle, the equation is simplified due to the radius being 1. To find a missing coordinate on the unit circle:

• Start with the unit circle equation: \( x^2 + y^2 = 1 \).
• If the x-coordinate is known, you can substitute it into the equation to solve for \( y^2 \).

Given that the x-coordinate of a point \(P\) is \(-\frac{2}{5}\), substituting this value gives:

• \(\left(-\frac{2}{5}\right)^2 + y^2 = 1\)
• Calculate \(\left(-\frac{2}{5}\right)^2\) to get \(\frac{4}{25}\).
• Substitute \( \frac{4}{25} \) back into the equation, resulting in \( \frac{4}{25} + y^2 = 1 \).

Finally, isolate \( y^2 \) to find \( y \) and determine the complete coordinates of point \(P\).

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that connects the sine and cosine of angles. However, it also emerges naturally when dealing with the unit circle. It states that for any point \((x, y)\) on the unit circle:\[x^2 + y^2 = 1\]This is the same as the equation of the unit circle, showing the deep connection between geometry and algebra. Here's how it applies:

• If you know one trigonometric value (like cosine) and it's on the unit circle, you can find the other (sine) using the identity.
• In the unit circle, cosine is the x-coordinate, and sine is the y-coordinate of a point \(P\).

Thus, given one coordinate, you can always solve for the other using the Pythagorean identity. This relationship helps greatly in problems regarding the unit circle and trigonometric functions, such as determining the coordinates from one known value, as shown in the solution for point \(P\) with \(x = -\frac{2}{5}\).