If The Point { \left(x, \frac{\sqrt{3}}{2}\right)$}$ Is On The Unit Circle, What Is { X$}$?A. { \frac{\sqrt{3}}{2}$}$ B. { \frac{2}{\sqrt{3}}$}$ C. { -\frac{\sqrt{3}}{2}$}$ D.
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 a crucial tool 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.
Understanding the Unit Circle
The unit circle is defined by the equation x^2 + y^2 = 1, where x and y are the coordinates of a point on the circle. The unit circle is symmetric about the x-axis and the y-axis, and it intersects the x-axis at the points (1, 0) and (-1, 0). The unit circle also intersects the y-axis at the points (0, 1) and (0, -1).
Finding x on the Unit Circle
To find the value of x on the unit circle given a point on the circle, we can use the equation of the unit circle. Let's consider the point (x, y) = (x, \frac{\sqrt{3}}{2}). We can substitute this point into the equation of the unit circle to solve for x.
Substituting the Point into the Equation
Substituting the point (x, y) = (x, \frac{\sqrt{3}}{2}) into the equation of the unit circle, we get:
x^2 + (\frac{\sqrt{3}}{2})^2 = 1
Simplifying the Equation
Simplifying the equation, we get:
x^2 + \frac{3}{4} = 1
Isolating x
Subtracting \frac{3}{4} from both sides of the equation, we get:
x^2 = 1 - \frac{3}{4}
x^2 = \frac{1}{4}
Taking the Square Root
Taking the square root of both sides of the equation, we get:
x = \pm \sqrt{\frac{1}{4}}
x = \pm \frac{1}{2}
Choosing the Correct Answer
Since the point (x, y) = (x, \frac{\sqrt{3}}{2}) is on the unit circle, we know that x must be positive. Therefore, the correct answer is x = \frac{1}{2}.
Conclusion
In this article, we have shown how to find the value of x on the unit circle given a point on the circle. We used the equation of the unit circle to substitute the point (x, y) = (x, \frac{\sqrt{3}}{2}) and solved for x. We found that x = \frac{1}{2}.
Answer
The correct answer is:
- A. {\frac{\sqrt{3}}{2}$}$ is incorrect
- B. {\frac{2}{\sqrt{3}}$}$ is incorrect
- C. {-\frac{\sqrt{3}}{2}$}$ is incorrect
- D. {\frac{1}{2}$}$ is correct
Unit Circle Q&A =====================
Frequently Asked Questions about the Unit Circle
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. In this article, we will answer some frequently asked questions about the unit circle.
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 of the coordinate plane, which is (0, 0).
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 use the equation of the unit circle. Substitute the point into the 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 using 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 use the equation of the unit circle. Substitute the point into the 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 a crucial tool 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.
Q: What are some common applications of the unit circle?
A: Some common applications of the unit circle include:
- Solving problems involving right triangles
- Solving problems involving circular functions
- Finding the values of the trigonometric functions
- Graphing trigonometric functions
Conclusion
In this article, we have answered some frequently asked questions about the unit circle. The unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is used to define the trigonometric functions and is a crucial tool in solving problems involving right triangles and circular functions.
Additional Resources
For more information about the unit circle, you can consult the following resources: