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Chapter 1: Problem 38

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

Short Answer

Expert verified
Yes, the point \((-1,0)\) lies on the unit circle as it satisfies the equation \(x^2 + y^2 = 1\).

Step by step solution

01

Recall the Unit Circle Equation

The equation of the unit circle is given by \(x^2 + y^2 = 1\). Any point \((x, y)\) that lies on the unit circle will satisfy this equation.
02

Substitute the Point into the Equation

We have the point \((-1,0)\). Substitute \(x = -1\) and \(y = 0\) into the equation of the unit circle to check if it satisfies the equation: \[(-1)^2 + 0^2 = 1\]
03

Simplify the Expression

Calculate the left side of the equation: \[1 + 0 = 1\] This simplifies to \(1\), which is equal to the right side of the equation \(1\). Thus, the equation \((-1)^2 + 0^2 = 1\) holds true.
04

Conclusion

Since the equation holds true \((1 = 1)\), the point \((-1,0)\) lies on the graph of the unit circle.

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Key Concepts

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

Equation of a Circle
At the heart of the unit circle lies its equation, a fundamental concept in trigonometry and coordinate geometry. The equation of the unit circle is expressed as \(x^2 + y^2 = 1\). This formula represents a circle centered at the origin \(0,0\) with a radius of 1.
  • The "unit" in a unit circle indicates that the radius is one unit. This standardized size is what makes the unit circle a particularly useful tool in various mathematical disciplines.
  • In the equation, \(x\) and \(y\) are any coordinates of points lying on the circle.
Understanding this equation is crucial as it forms the basis for analyzing the trigonometric functions of any angle positioned in the unit circle. It allows for easy trigonometric calculations and visualization of angles in standard position.
Coordinates
Coordinates are essential for determining the position of any point in a plane. In the context of the unit circle, coordinates refer to the \(x, y\) values that satisfy the unit circle equation \(x^2 + y^2 = 1\).
  • The \(x\) coordinate is the horizontal distance from the origin. A point with a negative \(x\) value is located to the left of the origin, like our example point \((-1,0)\).
  • The \(y\) coordinate represents the vertical distance from the origin. For the point \((-1, 0)\), the \(y\) value is zero, placing it exactly on the x-axis.
Using coordinates, one can visualize and verify a point's position not just in geometric shapes but also within algebraic contexts, like determining if a point truly lies on a given circle.
Verification of Points
Verification of points is a straightforward process of determining whether a given point satisfies a specific equation. For the unit circle, this means checking if a point's coordinates fulfill the equation \(x^2 + y^2 = 1\).Let's take the given point \((-1,0)\):- Substitute the point into the equation: Replace \(x = -1\) and \(y = 0\) into the formula.- Calculate: \((-1)^2 + 0^2 = 1\), which simplifies further to \(1 + 0 = 1\).Since the equation holds true \(1 = 1\), the point indeed lies on the unit circle. This verification process can be applied to any point and is foundational in ensuring mathematical accuracy in graphing or solving equations involving circles.

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Most popular questions from this chapter

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \cos \theta \tan \theta=\sin \theta $$

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