Solve The Equation:${ -\frac{\pi}{3} \cos \frac{\pi}{6} = \frac{1}{2} \left( \sin \frac{\pi}{2} + \sin \frac{\pi}{6} \right) }$Provide Your Answer Here:

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific trigonometric equation involving sine and cosine functions. We will break down the solution into manageable steps, making it easier for readers to understand and follow along.

The Given Equation

The given equation is:

${ -\frac{\pi}{3} \cos \frac{\pi}{6} = \frac{1}{2} \left( \sin \frac{\pi}{2} + \sin \frac{\pi}{6} \right) }$

This equation involves the cosine and sine functions, and we need to simplify it to find the solution.

Step 1: Simplify the Equation

To simplify the equation, we can start by evaluating the trigonometric functions at the given angles.

• ${\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}}$
• ${\sin \frac{\pi}{2} = 1}$
• ${\sin \frac{\pi}{6} = \frac{1}{2}}$

Substituting these values into the equation, we get:

${ -\frac{\pi}{3} \left( \frac{\sqrt{3}}{2} \right) = \frac{1}{2} \left( 1 + \frac{1}{2} \right) }$

Step 2: Simplify the Left-Hand Side

Now, let's simplify the left-hand side of the equation.

${ -\frac{\pi}{3} \left( \frac{\sqrt{3}}{2} \right) = -\frac{\pi \sqrt{3}}{6} }$

Step 3: Simplify the Right-Hand Side

Next, let's simplify the right-hand side of the equation.

${ \frac{1}{2} \left( 1 + \frac{1}{2} \right) = \frac{1}{2} \left( \frac{3}{2} \right) = \frac{3}{4} }$

Step 4: Equate the Two Sides

Now that we have simplified both sides of the equation, we can equate them.

${ -\frac{\pi \sqrt{3}}{6} = \frac{3}{4} }$

Step 5: Solve for the Unknown

To solve for the unknown, we can start by multiplying both sides of the equation by 24, which is the least common multiple of 6 and 4.

${ -4 \pi \sqrt{3} = 18 }$

Next, we can divide both sides of the equation by -4.

${ \pi \sqrt{3} = -\frac{18}{4} }$

${ \pi \sqrt{3} = -\frac{9}{2} }$

Conclusion

In this article, we solved a trigonometric equation involving sine and cosine functions. We broke down the solution into manageable steps, making it easier for readers to understand and follow along. By simplifying the equation and equating the two sides, we were able to solve for the unknown. This problem demonstrates the importance of trigonometric functions in mathematics and their applications in real-world problems.

Final Answer

The final answer is:

${ \boxed{-\frac{9}{2\pi\sqrt{3}}} }$

Note: The final answer is in the form of a fraction, and it is not possible to simplify it further.

Additional Resources

For more information on trigonometric equations and their solutions, please refer to the following resources:

• Trigonometric Equations
• Solving Trigonometric Equations
• Trigonometric Functions

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the properties of waves.

Q: What are the common trigonometric functions?

A: The common trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cotangent (cot)
• Secant (sec)
• Cosecant (csc)

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can follow these steps:

1. Simplify the equation by combining like terms and using trigonometric identities.
2. Isolate the trigonometric function by moving all other terms to one side of the equation.
3. Use trigonometric properties, such as the unit circle and trigonometric identities, to find the solution.
4. Check your solution by plugging it back into the original equation.

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 solve trigonometric equations?

A: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions and can be used to solve trigonometric equations.

• The x-coordinate of a point on the unit circle is equal to the cosine of the angle formed by the point and the positive x-axis.
• The y-coordinate of a point on the unit circle is equal to the sine of the angle formed by the point and the positive x-axis.

Q: What are some real-world applications of trigonometric equations?

A: Trigonometric equations have many real-world applications, including:

• Physics and Engineering: Trigonometric equations are used to model the motion of objects, the behavior of electrical circuits, and the properties of waves.
• Navigation: Trigonometric equations are used in navigation to determine the position and direction of objects.
• Computer Graphics: Trigonometric equations are used in computer graphics to create 3D models and animations.
• Medical Imaging: Trigonometric equations are used in medical imaging to reconstruct images of the body.

Q: How can I practice solving trigonometric equations?

A: There are many resources available to help you practice solving trigonometric equations, including:

• Textbooks and Workbooks: There are many textbooks and workbooks available that provide practice problems and examples of trigonometric equations.
• Online Resources: There are many online resources available, including websites and apps, that provide practice problems and examples of trigonometric equations.
• Practice Tests: You can take practice tests to assess your knowledge and skills in solving trigonometric equations.

Conclusion

Solving trigonometric equations is an important skill in mathematics and has many real-world applications. By understanding the common trigonometric functions, using trigonometric identities, and applying the unit circle, you can solve trigonometric equations and apply them to real-world problems. Remember to practice regularly to improve your skills and knowledge.