Select The Correct Answer.If The Point { \left(x, \frac{\sqrt{3}}{2}\right)$}$ Is On The Unit Circle, What Could The Value Of { X$}$ Be?A. { -\frac{\sqrt{3}}{2}$}$ B. { \frac{2}{\sqrt{3}}$}$ C.

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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 of an angle, and it plays a crucial role in solving problems involving right triangles and circular functions.

Understanding the Unit Circle

The unit circle is defined by the equation x2+y2=1x^2 + y^2 = 1, where xx and yy are the coordinates of a point on the circle. The unit circle is symmetric about the x-axis and the y-axis, and it has a radius of 1 unit. The points on the unit circle can be represented in polar coordinates, which are given by the radius and the angle formed by the positive x-axis and the line segment connecting the origin to the point.

The Given Point on the Unit Circle

The given point is (x,32)\left(x, \frac{\sqrt{3}}{2}\right). This point lies on the unit circle, which means that it satisfies the equation of the unit circle: x2+y2=1x^2 + y^2 = 1. We are asked to find the possible values of xx.

Using the Equation of the Unit Circle

We can substitute the given values of xx and yy into the equation of the unit circle to get:

x2+(32)2=1x^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1

Simplifying the equation, we get:

x2+34=1x^2 + \frac{3}{4} = 1

Subtracting 34\frac{3}{4} from both sides, we get:

x2=14x^2 = \frac{1}{4}

Taking the square root of both sides, we get:

x=±12x = \pm \frac{1}{2}

Evaluating the Possible Values of x

We have found two possible values of xx: 12\frac{1}{2} and −12-\frac{1}{2}. However, we need to check if these values satisfy the given point on the unit circle.

Checking the Values of x

We can substitute the values of xx into the equation of the unit circle to check if they satisfy the given point:

For x=12x = \frac{1}{2}:

(12)2+(32)2=14+34=1\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1

This satisfies the equation of the unit circle.

For x=−12x = -\frac{1}{2}:

(−12)2+(32)2=14+34=1\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1

This also satisfies the equation of the unit circle.

Conclusion

We have found two possible values of xx: 12\frac{1}{2} and −12-\frac{1}{2}. Both values satisfy the equation of the unit circle and the given point on the circle. Therefore, the value of xx could be either 12\frac{1}{2} or −12-\frac{1}{2}.

Answer

The correct answer is:

Q&A: Exploring 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 explore the unit circle and answer some common questions related to it.

Q: What is the equation of the unit circle?

A: The equation of the unit circle is x2+y2=1x^2 + y^2 = 1, where xx and yy 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 you represent points on the unit circle?

A: Points on the unit circle can be represented in polar coordinates, which are given by the radius and the angle formed by the positive x-axis and the line segment connecting the origin to the point.

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

A: The unit circle is used to define the trigonometric functions of an angle. The values of sine, cosine, and tangent can be determined by the coordinates of a point on the unit circle.

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

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

Q: What are some common points on the unit circle?

A: Some common points on the unit circle include (1, 0), (-1, 0), (0, 1), and (0, -1).

Q: How do you graph the unit circle?

A: To graph the unit circle, you can use a coordinate plane and plot the points that satisfy the equation of the unit circle.

Q: What are some real-world applications of the unit circle?

A: The unit circle has many real-world applications, including physics, engineering, and computer science. It is used to model periodic phenomena, such as the motion of a pendulum or the vibration of a spring.

Q: How do you use the unit circle to solve problems?

A: To use the unit circle to solve problems, you can use the coordinates of a point on the circle to determine the values of trigonometric functions, such as sine, cosine, and tangent.

Conclusion

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 have explored the unit circle and answered some common questions related to it.

Frequently Asked Questions

  • What is the equation of the unit circle?
  • What is the radius of the unit circle?
  • What is the center of the unit circle?
  • How do you represent points on the unit circle?
  • What is the relationship between the unit circle and trigonometric functions?
  • How do you find the value of x given a point on the unit circle?
  • What are some common points on the unit circle?
  • How do you graph the unit circle?
  • What are some real-world applications of the unit circle?
  • How do you use the unit circle to solve problems?

Answer Key

  • The equation of the unit circle is x2+y2=1x^2 + y^2 = 1.
  • The radius of the unit circle is 1 unit.
  • The center of the unit circle is the origin of the coordinate plane, which is (0, 0).
  • Points on the unit circle can be represented in polar coordinates.
  • The unit circle is used to define the trigonometric functions of an angle.
  • To find the value of x given a point on the unit circle, you can substitute the values of x and y into the equation of the unit circle and solve for x.
  • Some common points on the unit circle include (1, 0), (-1, 0), (0, 1), and (0, -1).
  • To graph the unit circle, you can use a coordinate plane and plot the points that satisfy the equation of the unit circle.
  • The unit circle has many real-world applications, including physics, engineering, and computer science.
  • To use the unit circle to solve problems, you can use the coordinates of a point on the circle to determine the values of trigonometric functions, such as sine, cosine, and tangent.