91Ó°ÊÓ

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric figures and their properties. The coordinate plane consists of two axes, the x-axis and the y-axis, which intersect at the origin point \(0,0\). Each point on the plane is represented by a pair of numbers, \(x\) and \(y\), known as coordinates. In the context of the unit circle, the point \(P(x,y)\) falls on a circle with a radius of 1 centered at the origin. This simply means that any point on the unit circle is exactly one unit away from the origin.

• The x-coordinate represents the horizontal distance from the origin.
• The y-coordinate represents the vertical distance.
• On the unit circle, these coordinates satisfy the equation \(x^2 + y^2 = 1\).

Understanding coordinate geometry allows us to analyze the position of points and the shapes they form on a graph. For the given problem, the x-coordinate of point \(P\) is \(\frac{4}{5}\) which we use in the equation of the unit circle to solve for the missing y-coordinate. Since the y-coordinate is described as positive, it narrows down our options for a precise solution.

Trigonometry is a field of mathematics focused on the study of triangles and the relationships between their angles and sides. In the context of a unit circle, trigonometry helps us understand the connection between angles and the coordinates of points on the circle. The unit circle in trigonometry is a circle with a radius of 1, centered at the origin of a coordinate system.

• The angle \(\theta\) in the unit circle can be understood as the angle between the positive x-axis and a line connecting the origin to a point \(P(x, y)\).
• Key trigonometric functions, such as sine and cosine, are defined using the coordinates of this point: \(\cos(\theta) = x\text{ and }\sin(\theta) = y\).
• The unit circle allows us to visualize these functions and understand their behavior as \(\theta\) changes.

In the problem, while we focused more on the coordinates, the connection implies that for \(x = \frac{4}{5}\), we could also say \(\cos(\theta) = \frac{4}{5}\). The positive \(y\)-coordinate found in the solution, \(\frac{3}{5}\), corresponds to \(\sin(\theta)\), confirming that it must be the sine of a positive angle. Thus, the trigonometry of the unit circle helps verify our computed results.

The equation of a circle is a fundamental concept in coordinate geometry. It provides a way to algebraically represent a circle in the coordinate plane. A generic circle's equation with center at \((h, k)\) and a radius \(r\) is given by:\[ (x-h)^2 + (y-k)^2 = r^2 \]The unit circle, however, is a special case where the center is at the origin \( (0,0)\) and the radius \(r\) is 1. Hence, the equation simplifies to:\[ x^2 + y^2 = 1 \]

• Any point \( (x, y)\) lying on this circle must satisfy the simplified equation.
• Solutions to this equation are \( x^2 + y^2 = 1 \).
• This implies that the sum of the squares of the coordinates \(x\) and \(y\) will always equal 1, indicating the consistent distance from the origin.

For the problem at hand, this equation enabled us to determine the y-coordinate when provided the x-coordinate of point \(P\). By solving this equation, we found that the positive y-coordinate satisfying it is \(y = \frac{3}{5}\). Understanding the equation of a circle is crucial to solving such problems, as it serves as the foundation for identifying coordinates of any point on a circle in algebraic terms.