Given $\cos (x) = \frac{1}{2}$, What Is The Value Of $\cos (x+\pi$\]?A. $\frac{\sqrt{3}}{2}$B. $-\frac{\sqrt{3}}{2}$C. $\frac{1}{2}$D. $-\frac{1}{2}$

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically the equation $\cos (x) = \frac{1}{2}$, and then find the value of $\cos (x+\pi)$.

Understanding the Given Equation

The given equation is $\cos (x) = \frac{1}{2}$. This equation represents a cosine function that has a value of $\frac{1}{2}$ at a certain angle $x$. To solve this equation, we need to find the values of $x$ that satisfy this condition.

Recalling the Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us visualize the relationships between the angles and the corresponding cosine and sine values. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.

Finding the Values of $x$

Using the unit circle, we can find the values of $x$ that satisfy the equation $\cos (x) = \frac{1}{2}$. The cosine function is positive in the first and fourth quadrants, and the value of $\frac{1}{2}$ corresponds to an angle of $60^\circ$ or $\frac{\pi}{3}$ radians in the first quadrant.

Using the Periodicity of the Cosine Function

The cosine function has a period of $2\pi$, which means that the function repeats itself every $2\pi$ radians. Therefore, if we add or subtract multiples of $2\pi$ to the angle $x$, we will still get the same value of $\cos (x)$.

Finding the Value of $\cos (x+\pi)$

Now that we have found the values of $x$ that satisfy the equation $\cos (x) = \frac{1}{2}$, we can find the value of $\cos (x+\pi)$. To do this, we need to use the angle addition formula for cosine, which states that $\cos (a+b) = \cos a \cos b - \sin a \sin b$.

Applying the Angle Addition Formula

Using the angle addition formula, we can write $\cos (x+\pi) = \cos x \cos \pi - \sin x \sin \pi$. Since $\cos \pi = -1$ and $\sin \pi = 0$, we can simplify the expression to $\cos (x+\pi) = -\cos x$.

Substituting the Value of $\cos x$

We know that $\cos x = \frac{1}{2}$, so we can substitute this value into the expression $\cos (x+\pi) = -\cos x$ to get $\cos (x+\pi) = -\frac{1}{2}$.

Conclusion

In this article, we have solved the trigonometric equation $\cos (x) = \frac{1}{2}$ and found the value of $\cos (x+\pi)$. We have used the unit circle, the periodicity of the cosine function, and the angle addition formula to find the solution. The final answer is $\boxed{-\frac{1}{2}}$.

Answer Key

A. $\frac{\sqrt{3}}{2}$ (Incorrect) B. $-\frac{\sqrt{3}}{2}$ (Incorrect) C. $\frac{1}{2}$ (Incorrect) D. $-\frac{1}{2}$ (Correct)

Additional Resources

For more information on trigonometry and solving trigonometric equations, please refer to the following resources:

• Trigonometry Wikipedia Article
• Solving Trigonometric Equations
• Unit Circle

Final Thoughts

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some frequently asked questions about trigonometry, specifically about solving equations and finding values.

Q: What is the unit circle?

A: The unit circle is a fundamental concept in trigonometry that helps us visualize the relationships between the angles and the corresponding cosine and sine values. It is a circle with a radius of 1, centered at the origin of a coordinate plane.

Q: How do I find the values of $x$ that satisfy the equation $\cos (x) = \frac{1}{2}$?

A: To find the values of $x$ that satisfy the equation $\cos (x) = \frac{1}{2}$, we can use the unit circle. The cosine function is positive in the first and fourth quadrants, and the value of $\frac{1}{2}$ corresponds to an angle of $60^\circ$ or $\frac{\pi}{3}$ radians in the first quadrant.

Q: What is the periodicity of the cosine function?

A: The cosine function has a period of $2\pi$, which means that the function repeats itself every $2\pi$ radians. Therefore, if we add or subtract multiples of $2\pi$ to the angle $x$, we will still get the same value of $\cos (x)$.

Q: How do I find the value of $\cos (x+\pi)$?

A: To find the value of $\cos (x+\pi)$, we can use the angle addition formula for cosine, which states that $\cos (a+b) = \cos a \cos b - \sin a \sin b$. Since $\cos \pi = -1$ and $\sin \pi = 0$, we can simplify the expression to $\cos (x+\pi) = -\cos x$.

Q: What is the final answer to the equation $\cos (x) = \frac{1}{2}$?

A: The final answer to the equation $\cos (x) = \frac{1}{2}$ is $\boxed{-\frac{1}{2}}$.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$
• $\sec x = \frac{1}{\cos x}$
• $\csc x = \frac{1}{\sin x}$

Q: How do I use the unit circle to find the values of sine and cosine?

A: To use the unit circle to find the values of sine and cosine, we can follow these steps:

1. Draw a diagram of the unit circle.
2. Identify the quadrant in which the angle lies.
3. Find the corresponding sine and cosine values using the unit circle.

Q: What are some real-world applications of trigonometry?

A: Some real-world applications of trigonometry include:

• Navigation: Trigonometry is used in navigation to calculate distances and directions.
• Physics: Trigonometry is used in physics to describe the motion of objects.
• Engineering: Trigonometry is used in engineering to design and build structures.
• Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we have answered some frequently asked questions about trigonometry, specifically about solving equations and finding values. We have covered topics such as the unit circle, periodicity, and angle addition formulas. We have also discussed some common trigonometric identities and real-world applications of trigonometry.