Problem 4 - Precalculus Mathematics for Calculus

Question

Show that the point is on the unit circle. $$\left(-\frac{5}{13}, \frac{12}{13}\right)$$

Step-by-Step Solution

Answer

The point is on the unit circle because the sum of the squares of its coordinates equals 1.

Step 1: Understand the Unit Circle Equation

The equation of a unit circle centered at the origin \(0,0\) is given by \(x^2 + y^2 = 1\). A point is on the unit circle if the sum of the squares of its coordinates equals 1.

Step 2: Substitute the Coordinates into the Equation

Substitute the x-coordinate \(-\frac{5}{13}\) and the y-coordinate \(\frac{12}{13}\) into the equation of the unit circle: \[\left(-\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 = 1\]

Step 3: Calculate the Squares of the Coordinates

Calculate \(-\frac{5}{13}\)^2 = \(\frac{25}{169}\) and \(\left(\frac{12}{13}\right)^2 = \frac{144}{169}\). These calculations give the squares of the x and y coordinates respectively.

Step 4: Add the Squared Values

Add the squared values obtained in the previous step: \[\frac{25}{169} + \frac{144}{169} = \frac{169}{169} = 1\]

Step 5: Conclusion

Since the sum of squares \(\frac{25}{169} + \frac{144}{169}\) equals 1, the point \left(-\frac{5}{13}, \frac{12}{13}\right)\ is on the unit circle as required by the condition.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes in a coordinate plane. In this system, every point is defined by coordinates, usually in the form of (x, y).

The great advantage of coordinate geometry is that it allows you to illustrate geometric shapes and solve problems using algebraic equations. For example, the location of a point can be described with its coordinates, and equations can be used to express and analyze the spatial properties and relationships of geometric figures.

When talking about circles, specifically the unit circle, we can use coordinate geometry to describe the positions of points along its circumference.

Pythagorean Identity

The Pythagorean Identity is a vital concept in trigonometry that arises from the Pythagorean theorem. The theorem states that for a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

For the unit circle, this identity takes the form:

• The equation: \( x^2 + y^2 = 1 \)
• Here, the radius of the unit circle equals 1, similar to the hypotenuse of a right triangle.

In the context of the exercise, substituting

• the x-coordinate \(-\frac{5}{13}\)
• and the y-coordinate \(\frac{12}{13}\)

yields the Pythagorean identity of the unit circle. This identity confirms that these specific coordinate values satisfy the equation of the circle, which makes it a rigorous tool for determining whether points are on the circle.

Equation of a Circle

The equation of a circle is a fundamental concept in coordinate geometry. The standard form for a circle centered at the origin is:

• \( x^2 + y^2 = r^2 \), where \( r \) is the radius.

The unit circle, specifically, has a radius \( r = 1 \), so its equation simplifies to \( x^2 + y^2 = 1 \).

In solving problems involving the unit circle, such as confirming a point lies on it, we substitute the point's coordinates into this equation. If the sum of the squares equals 1, as it does with the point

• \( \left(-\frac{5}{13}, \frac{12}{13}\right) \)

, it confirms the point is on the circle.

This method is powerful because it turns potentially complex geometric concepts into straightforward algebraic manipulations. This simplicity is one of the reasons the unit circle is a cornerstone in various branches of mathematics and its applications.