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Chapter 5: Problem 2

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

Short Answer

Expert verified
The point is on the unit circle.

Step by step solution

01

Understanding the Problem

We need to verify if the given point \(\left(-\frac{5}{13}, \frac{12}{13}\right)\) is on the unit circle. A point is on the unit circle if it satisfies the equation \(x^2 + y^2 = 1\), where \(x\) and \(y\) are coordinates of the point.
02

Substitute the Point's Coordinates

Substitute \(x = -\frac{5}{13}\) and \(y = \frac{12}{13}\) into the equation \(x^2 + y^2 = 1\). This becomes:\[ \left(-\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 = 1.\]
03

Calculate the Squares

Compute the squares of each coordinate:1. \(\left(-\frac{5}{13}\right)^2 = \frac{25}{169}\).2. \(\left(\frac{12}{13}\right)^2 = \frac{144}{169}\).
04

Add the Squares

Add the squared values:\[ \frac{25}{169} + \frac{144}{169} = \frac{169}{169} = 1.\] This shows that \(x^2 + y^2 = 1\), confirming that the point is on the unit circle.
05

Conclusion

Since the sum of the squares of the coordinates equals 1, the point \(\left(-\frac{5}{13}, \frac{12}{13}\right)\) is indeed on the unit circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that connects geometry and algebra. It allows us to describe geometrical shapes and figures using algebraic equations and coordinates. This involves the use of a coordinate system, where each point on a plane is defined by an ordered pair \( (x, y) \).
  • These coordinates tell us the position of the point in the plane.
  • Coordinate geometry combines the number line with a plane, called the Cartesian plane.
A Cartesian plane is divided by two axes:
  • The x-axis, which is horizontal.
  • The y-axis, which is vertical.
Proper understanding of these axes and coordinate points is essential, especially when dealing with objects like the unit circle, which uses the formula \( x^2 + y^2 = 1 \). This formula helps us find or verify coordinates that lay on the circle.
Unit Circle Equation
The unit circle is a fundamental concept in mathematics, especially in trigonometry. It is named 'unit' because its radius is 1. The equation of the unit circle is \( x^2 + y^2 = 1 \).
This equation is derived from the Pythagorean theorem, where:
  • \(x\) and \(y\) are the horizontal and vertical distances from the origin (0,0) to any point (x,y) on the circle.
  • Since the radius is 1, these distances must satisfy the equation such that when squared and summed, they equal 1.
For example, to verify if a point is on the unit circle, we use its equation by substituting the x and y values:
  • If \(x^2 + y^2 = 1\), the point is on the circle.
  • Otherwise, it is outside the circle or on a different circle.
This principle is vital in determining positions on the unit circle and has applications in defining sine and cosine functions in trigonometry.
Verification Process
The verification process is a straightforward yet crucial method to confirm whether a point lies on the unit circle. This process involves a few simple steps:
**Substitute the Coordinates**
  • Take the given point's coordinates, say \( (x, y) \), and plug them into the equation \( x^2 + y^2 = 1 \).
**Square the Coordinates**
  • Calculate the square of each coordinate to see how much they contribute to the equation.
**Add the Squares**
  • Sum the results obtained from squaring to check if the result is 1.
If the sum equals 1, the point is on the unit circle, confirming its validity. This process is crucial because it simplifies tasks such as determining angles and distances around the circle. It's an effective way to assure mathematical accuracy and understanding of positional relationships in coordinate geometry.

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