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Chapter 1: Problem 39

Verify that each point lies on the graph of the unit circle. \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)

Short Answer

Expert verified
The point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) is on the unit circle.

Step by step solution

01

Recall the Unit Circle Equation

The unit circle is defined by the equation \(x^2 + y^2 = 1\). To determine if a point lies on the unit circle, its coordinates \((x, y)\) must satisfy this equation.
02

Substitute Coordinates into the Equation

For the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), let \(x = \frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\). Substitute these values into the unit circle equation: \(\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1\).
03

Calculate \(x^2\) and \(y^2\)

Calculate \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) and \(\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).
04

Add the Results

Add \(\frac{1}{4}\) and \(\frac{3}{4}\) to get \(\frac{1}{4} + \frac{3}{4} = 1\).
05

Conclusion

Since \(x^2 + y^2 = 1\), the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) satisfies the unit circle equation, confirming that it 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.

Verifying Coordinates on the Unit Circle
Understanding how to verify if a point lies on a unit circle begins with recognizing the equation of the unit circle, which is given by \(x^2 + y^2 = 1\). This equation represents a circle centered at the origin \((0, 0)\) with a radius of 1. The coordinates of any point on this circle must satisfy this equation. To verify if a point, such as \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), lies on the unit circle, simply substitute the \(x\) and \(y\) values into the equation. If the equation holds true, the point indeed lies on the circle.For example:
  • Substitute \(x = \frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\) into the unit circle equation.
  • Calculate \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
  • Calculate \(\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).
  • Add the results: \(\frac{1}{4} + \frac{3}{4} = 1\).
Since the equation is satisfied, the point is verified to be on the unit circle.
Substituting Values in Unit Circle Equation
Equation substitution is a fundamental algebraic skill that allows you to determine whether a point is part of a specific curve or line. When dealing with the unit circle, you need to substitute the given coordinates into the unit circle equation \(x^2 + y^2 = 1\).Here's how you do it:
  • Identify the given point, such as \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
  • Recognize the correlation, where \(x = \frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\).
  • Plug these values into the equation.
  • Check to see if the left side equals 1.
Performing this substitution helps confirm whether or not the point lies on the unit circle. It's like placing a puzzle piece to see if it fits correctly. If the sides do not equal, the point is not on the circle.
Understanding the Pythagorean Identity
The Pythagorean identity is a key concept in understanding trigonometry and unit circles. It derives from the Pythagorean theorem and is expressed in the form of the equation \(x^2 + y^2 = 1\) for a unit circle. This identity tells us that for any point on a circle with radius 1, the square of its \(x\)-coordinate plus the square of its \(y\)-coordinate equals 1. This is crucial because:
  • Every point on the unit circle adheres to this identity.
  • The identity forms a foundational basis for understanding trigonometric ratios since sin and cos are essentially the post-angles coordinates of a point on the circle.
  • It seamlessly ties geometry with algebraic principles.
By using the Pythagorean identity, mathematicians can easily verify or predict the presence of points on the unit circle, providing both a powerful check and a useful computational tool.

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Most popular questions from this chapter

Write each of the following in terms of \(\sin \theta\) and \(\cos \theta\); then simplify if possible: \(\tan \theta+\sec \theta\)

Show that each of the following statements is an identity by transforming the left side of each one into the right side. \(\tan ^{2} \theta+1=\sec ^{2} \theta\)

Multiply. \((2 \cos \theta+3)(4 \cos \theta-5)\)

Find an angle \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) for which \(\cos \theta=-1\).

Write each of the following in terms of \(\sin \theta\) only: \(\sec \theta\)
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