What Is The Exact Value Of The Expression?$\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right$\]A. $\frac{4 \sqrt{3} - \sqrt{6}}{4}$B. $\frac{4 \sqrt{3} +

Introduction

In this article, we will explore the exact value of a given trigonometric expression involving tangent, cosine, and sine functions. The expression is $\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right)$. We will use various trigonometric identities and properties to simplify the expression and find its exact value.

Understanding the Expression

The given expression involves three trigonometric functions: tangent, cosine, and sine. The tangent function is defined as the ratio of the sine and cosine functions, i.e., $\tan(x) = \frac{\sin(x)}{\cos(x)}$. The cosine and sine functions are periodic with a period of $2\pi$, and their values repeat every $2\pi$ radians.

Simplifying the Expression

To simplify the expression, we can use the following trigonometric identities:

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$
• $\cos(x) = \cos(-x)$
• $\sin(x) = -\sin(-x)$

Using these identities, we can rewrite the expression as:

$\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right)$

$= \frac{\sin \left(\frac{\pi}{3}\right)}{\cos \left(\frac{\pi}{3}\right)} + \cos \left(\frac{5 \pi}{6}\right) \cdot (-\sin \left(\frac{3 \pi}{4}\right))$

$= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

$= \sqrt{3} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

Evaluating the Cosine and Sine Functions

To evaluate the cosine and sine functions, we can use the following values:

• $\cos \left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2}$
• $\sin \left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}$

Substituting these values into the expression, we get:

$= \sqrt{3} - (-\frac{\sqrt{3}}{2}) \cdot \frac{\sqrt{2}}{2}$

$= \sqrt{3} + \frac{\sqrt{3} \cdot \sqrt{2}}{4}$

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

Simplifying the Expression Further

To simplify the expression further, we can combine the terms:

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

$= \frac{4\sqrt{3} + \sqrt{6}}{4}$

Conclusion

In this article, we have explored the exact value of the given trigonometric expression involving tangent, cosine, and sine functions. We have used various trigonometric identities and properties to simplify the expression and find its exact value. The final answer is $\frac{4\sqrt{3} + \sqrt{6}}{4}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Discussion

The given expression involves three trigonometric functions: tangent, cosine, and sine. The tangent function is defined as the ratio of the sine and cosine functions, i.e., $\tan(x) = \frac{\sin(x)}{\cos(x)}$. The cosine and sine functions are periodic with a period of $2\pi$, and their values repeat every $2\pi$ radians.

The expression can be simplified using various trigonometric identities and properties. We have used the following identities:

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$
• $\cos(x) = \cos(-x)$
• $\sin(x) = -\sin(-x)$

Using these identities, we can rewrite the expression as:

$\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right)$

$= \frac{\sin \left(\frac{\pi}{3}\right)}{\cos \left(\frac{\pi}{3}\right)} + \cos \left(\frac{5 \pi}{6}\right) \cdot (-\sin \left(\frac{3 \pi}{4}\right))$

$= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

$= \sqrt{3} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

To evaluate the cosine and sine functions, we can use the following values:

• $\cos \left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2}$
• $\sin \left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}$

Substituting these values into the expression, we get:

$= \sqrt{3} - (-\frac{\sqrt{3}}{2}) \cdot \frac{\sqrt{2}}{2}$

$= \sqrt{3} + \frac{\sqrt{3} \cdot \sqrt{2}}{4}$

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

To simplify the expression further, we can combine the terms:

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

$= \frac{4\sqrt{3} + \sqrt{6}}{4}$

$$

Q: What is the exact value of the expression $\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right)$?

A: The exact value of the expression is $\frac{4\sqrt{3} + \sqrt{6}}{4}$.

Q: How did you simplify the expression?

A: We used various trigonometric identities and properties to simplify the expression. We started by rewriting the expression using the tangent, cosine, and sine functions. Then, we used the following identities:

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$
• $\cos(x) = \cos(-x)$
• $\sin(x) = -\sin(-x)$

Using these identities, we were able to rewrite the expression as:

$\tan \left(\frac{\pi}{3}\right) + \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(-\frac{3 \pi}{4}\right)$

$= \frac{\sin \left(\frac{\pi}{3}\right)}{\cos \left(\frac{\pi}{3}\right)} + \cos \left(\frac{5 \pi}{6}\right) \cdot (-\sin \left(\frac{3 \pi}{4}\right))$

$= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

$= \sqrt{3} - \cos \left(\frac{5 \pi}{6}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$

Q: How did you evaluate the cosine and sine functions?

A: We used the following values:

• $\cos \left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2}$
• $\sin \left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}$

Substituting these values into the expression, we got:

$= \sqrt{3} - (-\frac{\sqrt{3}}{2}) \cdot \frac{\sqrt{2}}{2}$

$= \sqrt{3} + \frac{\sqrt{3} \cdot \sqrt{2}}{4}$

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

Q: Can you explain the final step of simplifying the expression?

A: Yes, the final step of simplifying the expression was to combine the terms:

$= \sqrt{3} + \frac{\sqrt{6}}{4}$

$= \frac{4\sqrt{3} + \sqrt{6}}{4}$

This is the final answer to the expression.

Q: What are some common trigonometric identities that can be used to simplify expressions?

A: Some common trigonometric identities that can be used to simplify expressions include:

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$
• $\cos(x) = \cos(-x)$
• $\sin(x) = -\sin(-x)$
• $\sin^2(x) + \cos^2(x) = 1$
• $\tan(x) = \frac{\sin(x)}{\cos(x)}$

These identities can be used to rewrite expressions and simplify them.

Q: How can I apply these trigonometric identities to simplify expressions?

A: To apply these trigonometric identities to simplify expressions, you can follow these steps:

1. Identify the trigonometric functions in the expression.
2. Use the identities to rewrite the expression.
3. Simplify the expression using algebraic manipulations.
4. Check the final answer to ensure it is correct.

By following these steps, you can use trigonometric identities to simplify expressions and find their exact values.