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Chapter 1: Problem 37

Verify that each point lies on the graph of the unit circle. \((0,-1)\)

Short Answer

Expert verified
Point \((0,-1)\) lies on the unit circle.

Step by step solution

01

Recall the Equation of the Unit Circle

The unit circle is centered at the origin \((0, 0)\) and has a radius of 1. The standard equation for the unit circle is: \[ x^2 + y^2 = 1 \]
02

Substitute the Point into the Equation

Take the point \((0, -1)\) and substitute the coordinates into the unit circle's equation. This means setting \(x = 0\) and \(y = -1\): \[ 0^2 + (-1)^2 = 1 \]
03

Simplify the Equation

Calculate each term separately and combine them: - \(0^2 = 0\)- \((-1)^2 = 1\)Thus, the equation becomes: \[ 0 + 1 = 1 \]
04

Verify the Equality

Now, verify that the left side of the equation equals the right side: \[ 1 = 1 \] Since both sides are equal, the point \((0, -1)\) satisfies the unit circle equation.

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Key Concepts

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

Equation of the Unit Circle
The unit circle is a fundamental concept in trigonometry and geometry. It is a circle with a radius of one unit, centered at the origin \(0, 0\). These properties lead directly to the standard equation of the unit circle: \[ x^2 + y^2 = 1 \]
  • Here, \(x\) and \(y\) are the coordinates of any point on the circle.
  • Since the radius is always 1, the equation holds true for any point on this circle.
This equation essentially states that the sum of the squares of the x-coordinate and y-coordinate of any point on the unit circle will always equal 1. This relationship makes it especially useful when dealing with trigonometric functions, as the unit circle can help visualize the sine, cosine, and tangent functions with ease.
Graphing
Graphing the unit circle is a straightforward way to visually understand and verify points. Since it is centered at the origin with a radius of 1, the unit circle touches the x-axis at 1 and -1, and the y-axis at 1 and -1.
  • To plot the unit circle, simply draw a circle with a center point at \(0, 0\) and a radius extending to any direction equal to 1 unit.
  • This uniform distance in every direction from \(0, 0\) creates the perfect circle.
  • Common points include (1, 0), (0, 1), (-1, 0), and (0, -1).
By understanding these aspects, checking if any point lies on the unit circle becomes a question of matching these coordinates into the circle's equation and verifying if they satisfy it.
Substitution Method
The substitution method is a powerful tool to verify whether a given point lies on the unit circle. This involves plugging the x and y values of the point into the equation \(x^2 + y^2 = 1\). Here's a step-by-step breakdown:
  • First, take the coordinates of the given point. For example, \(0, -1\).
  • Substitute these values into the equation. Set \(x = 0\) and \(y = -1\).
  • Calculate each part: \(0^2 = 0\) and \((-1)^2 = 1\).
  • Add these results: \(0 + 1 = 1\).
After substitution, check if the left side of the equation equals the right side. If they match, the point indeed lies on the unit circle. This straightforward approach simplifies the process of verifying points, making the equation of the unit circle a practical tool in geometry.

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