Which Of The Following Trigonometric Expressions Is Equivalent To The { X$}$ Coordinate Of The Terminal Point { \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$?A. { \cos \left(\frac{\pi}{3}\right)$}$ B. [$\sin
Understanding Trigonometric Expressions
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the concept of trigonometric expressions and determine which of the given expressions is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
The Basics of Trigonometry
Trigonometry involves the study of triangles, particularly right-angled triangles. The trigonometric functions, such as sine, cosine, and tangent, are used to describe the relationships between the sides and angles of these triangles. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
The Given Trigonometric Expressions
We are given two trigonometric expressions: {\cos \left(\frac{\pi}{3}\right)$}$ and {\sin \left(\frac{\pi}{3}\right)$}$. We need to determine which of these expressions is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
Analyzing the Given Point
The given point is {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$. This point lies in the first quadrant of the coordinate plane, and the {x$}$ coordinate is {\frac{\sqrt{2}}{2}$}$. We need to determine which of the given trigonometric expressions is equivalent to this value.
Evaluating the Trigonometric Expressions
To evaluate the trigonometric expressions, we need to consider the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The trigonometric functions can be evaluated by considering the coordinates of the points on the unit circle.
The Cosine Function
The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In the case of the unit circle, the cosine function can be evaluated by considering the {x$}$ coordinate of the point on the unit circle. The cosine function is given by:
{\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$}$
The Sine Function
The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In the case of the unit circle, the sine function can be evaluated by considering the {y$}$ coordinate of the point on the unit circle. The sine function is given by:
{\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$}$
Evaluating the Given Expressions
We need to evaluate the given expressions {\cos \left(\frac{\pi}{3}\right)$}$ and {\sin \left(\frac{\pi}{3}\right)$}$ to determine which of them is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
The Value of Cosine
The value of cosine at {\frac{\pi}{3}$}$ is given by:
{\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$}$
The Value of Sine
The value of sine at {\frac{\pi}{3}$}$ is given by:
{\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$}$
Determining the Equivalent Expression
We need to determine which of the given expressions is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$. The {x$}$ coordinate is {\frac{\sqrt{2}}{2}$}$, which is equal to the value of cosine at {\frac{\pi}{3}$}$.
Conclusion
In conclusion, the trigonometric expression {\cos \left(\frac{\pi}{3}\right)$}$ is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$. This is because the value of cosine at {\frac{\pi}{3}$}$ is {\frac{1}{2}$}$, which is equal to the {x$}$ coordinate of the given point.
Final Answer
The final answer is A. {\cos \left(\frac{\pi}{3}\right)$}$.
Understanding Trigonometric Expressions
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the concept of trigonometric expressions and determine which of the given expressions is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
Frequently Asked Questions
Q: What is the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$?
A: The {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$ is {\frac{\sqrt{2}}{2}$}$.
Q: Which of the given trigonometric expressions is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$?
A: The trigonometric expression {\cos \left(\frac{\pi}{3}\right)$}$ is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
Q: Why is the cosine function used to evaluate the {x$}$ coordinate of the terminal point?
A: The cosine function is used to evaluate the {x$}$ coordinate of the terminal point because it is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In the case of the unit circle, the cosine function can be evaluated by considering the {x$}$ coordinate of the point on the unit circle.
Q: What is the value of cosine at {\frac{\pi}{3}$}$?
A: The value of cosine at {\frac{\pi}{3}$}$ is {\frac{1}{2}$}$.
Q: How does the value of cosine at {\frac{\pi}{3}$}$ relate to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$?
A: The value of cosine at {\frac{\pi}{3}$}$ is equal to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$.
Q: What is the final answer to the problem?
A: The final answer is A. {\cos \left(\frac{\pi}{3}\right)$}$.
Conclusion
In conclusion, the trigonometric expression {\cos \left(\frac{\pi}{3}\right)$}$ is equivalent to the {x$}$ coordinate of the terminal point {\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$}$. This is because the value of cosine at {\frac{\pi}{3}$}$ is {\frac{1}{2}$}$, which is equal to the {x$}$ coordinate of the given point.
Final Answer
The final answer is A. {\cos \left(\frac{\pi}{3}\right)$}$.
Additional Resources
For more information on trigonometric expressions and the {x$}$ coordinate of the terminal point, please refer to the following resources:
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