Given Point on the Unit Circle

Oct 13, 2025 by recoveryshake.com 142 views

Introduction

The unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The unit circle is used to define the trigonometric functions, such as sine, cosine, and tangent, and is essential in solving problems involving right triangles and circular functions. In this article, we will explore how to find the value of x on the unit circle given a point on the circle.

The Unit Circle Equation

The equation of the unit circle is x^2 + y^2 = 1, where x and y are the coordinates of a point on the circle. This equation represents a circle with a radius of 1 unit, centered at the origin (0, 0).

Given Point on the Unit Circle

We are given a point on the unit circle: $\left(x, \frac{\sqrt{3}}{2}\right)$ . Our goal is to find the value of x.

Using the Unit Circle Equation

We can substitute the given point into the unit circle equation to solve for x:

x^2 + $\left(\frac{\sqrt{3}}{2}\right)^2$ = 1

Simplifying the Equation

We can simplify the equation by evaluating the square of $\frac{\sqrt{3}}{2}$ :

x^2 + $\frac{3}{4}$ = 1

Isolating x

To isolate x, we can subtract $\frac{3}{4}$ from both sides of the equation:

x^2 = 1 - $\frac{3}{4}$

x^2 = $\frac{1}{4}$

Taking the Square Root

We can take the square root of both sides of the equation to solve for x:

x = $\pm\sqrt{\frac{1}{4}}$

x = $\pm\frac{1}{2}$

Conclusion

We have found the value of x on the unit circle given a point on the circle. The value of x is $\pm\frac{1}{2}$ . However, we need to consider the context of the problem to determine the correct answer.

Answer Choice Analysis

Let's analyze the answer choices:

A. $\frac{1}{2}$ B. $\frac{\sqrt{3}}{2}$ C. $-\frac{\sqrt{3}}{2}$ D. $\frac{2}{\sqrt{3}}$

We can see that answer choice A is a possible solution, but we need to consider the context of the problem to determine if it is the correct answer.

Contextual Analysis

The problem states that the point $\left(x, \frac{\sqrt{3}}{2}\right)$ is on the unit circle. This means that the y-coordinate of the point is $\frac{\sqrt{3}}{2}$ , which is a known value on the unit circle. The x-coordinate of the point is what we are trying to find.

Conclusion

Based on the analysis, we can conclude that the value of x on the unit circle given a point on the circle is $\frac{1}{2}$ . This is because the point $\left(x, \frac{\sqrt{3}}{2}\right)$ is on the unit circle, and the x-coordinate of the point is $\frac{1}{2}$ .

Final Answer

$The final answer is $\boxed{\frac{1}{2}}$.<br/> **Unit Circle Q&A**$

Introduction

Q: What is the equation of the unit circle?

A: The equation of the unit circle is x^2 + y^2 = 1, where x and y are the coordinates of a point on the circle.

Q: What is the radius of the unit circle?

A: The radius of the unit circle is 1 unit.

Q: What is the center of the unit circle?

A: The center of the unit circle is the origin (0, 0) of the coordinate plane.

Q: How do I find the value of x on the unit circle given a point on the circle?

A: To find the value of x on the unit circle given a point on the circle, you can substitute the given point into the unit circle equation and solve for x.

Q: What is the relationship between the unit circle and the trigonometric functions?

A: The unit circle is used to define the trigonometric functions, such as sine, cosine, and tangent. The values of these functions can be found by evaluating the coordinates of a point on the unit circle.

Q: How do I find the value of y on the unit circle given a point on the circle?

A: To find the value of y on the unit circle given a point on the circle, you can substitute the given point into the unit circle equation and solve for y.

Q: What is the significance of the unit circle in mathematics?

A: The unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is used to define the trigonometric functions and is essential in solving problems involving right triangles and circular functions.

Q: Can I use the unit circle to solve problems involving right triangles?

A: Yes, you can use the unit circle to solve problems involving right triangles. The unit circle can be used to find the values of the trigonometric functions, which can then be used to solve problems involving right triangles.

Q: Can I use the unit circle to solve problems involving circular functions?

A: Yes, you can use the unit circle to solve problems involving circular functions. The unit circle can be used to find the values of the trigonometric functions, which can then be used to solve problems involving circular functions.

Conclusion

Final Answer

The final answer is that the unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is used to define the trigonometric functions and is essential in solving problems involving right triangles and circular functions.

Additional Resources

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