Evaluate The Expression:${ \frac{\sqrt{6} \cdot \operatorname{tg} 30^{\circ} \cdot \operatorname{sen} 45^{\circ}}{\csc 30^{\circ}+\sec 60^{\circ} \cdot \operatorname{ctg} 45^{\circ}} }$

Introduction

Trigonometric functions are a fundamental aspect of mathematics, and their applications can be seen in various fields, including physics, engineering, and navigation. In this article, we will delve into the evaluation of a complex expression involving trigonometric functions, specifically the tangent, sine, cosine, cosecant, secant, and cotangent. The expression is given as:

$$\frac{\sqrt{6} \cdot \operatorname{tg} 30^{\circ} \cdot \operatorname{sen} 45^{\circ}}{\csc 30^{\circ}+\sec 60^{\circ} \cdot \operatorname{ctg} 45^{\circ}}$$

Understanding the Trigonometric Functions Involved

Before we proceed with the evaluation, it is essential to understand the trigonometric functions involved in the expression. The functions are:

• Tangent (tg): The ratio of the sine of an angle to the cosine of the same angle.
• Sine (sen): The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
• Cosecant (csc): The reciprocal of the sine function.
• Secant (sec): The reciprocal of the cosine function.
• Cotangent (ctg): The reciprocal of the tangent function.

Evaluating the Expression

To evaluate the expression, we need to simplify the numerator and denominator separately.

Evaluating the Numerator

The numerator of the expression is given by:

$$\sqrt{6} \cdot \operatorname{tg} 30^{\circ} \cdot \operatorname{sen} 45^{\circ}$$

We know that:

• $\operatorname{tg} 30^{\circ} = \frac{1}{\sqrt{3}}$
• $\operatorname{sen} 45^{\circ} = \frac{1}{\sqrt{2}}$

Substituting these values, we get:

$$\sqrt{6} \cdot \frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{2}}$$

Simplifying further, we get:

$$\frac{\sqrt{6}}{\sqrt{3} \cdot \sqrt{2}}$$

Evaluating the Denominator

The denominator of the expression is given by:

$$\csc 30^{\circ}+\sec 60^{\circ} \cdot \operatorname{ctg} 45^{\circ}$$

We know that:

• $\csc 30^{\circ} = 2$
• $\sec 60^{\circ} = 2$
• $\operatorname{ctg} 45^{\circ} = 1$

Substituting these values, we get:

$$2 + 2 \cdot 1$$

Simplifying further, we get:

$$4$$

Simplifying the Expression

Now that we have evaluated the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

$$\frac{\frac{\sqrt{6}}{\sqrt{3} \cdot \sqrt{2}}}{4}$$

Simplifying further, we get:

$$\frac{\sqrt{6}}{4 \cdot \sqrt{3} \cdot \sqrt{2}}$$

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of the denominator is:

$$4 \cdot \sqrt{3} \cdot \sqrt{2}$$

Multiplying the numerator and denominator by the conjugate, we get:

$$\frac{\sqrt{6} \cdot 4 \cdot \sqrt{3} \cdot \sqrt{2}}{(4 \cdot \sqrt{3} \cdot \sqrt{2})^2}$$

Simplifying further, we get:

$$\frac{4 \cdot \sqrt{6} \cdot \sqrt{3} \cdot \sqrt{2}}{16 \cdot 3 \cdot 2}$$

Final Simplification

Simplifying further, we get:

$$\frac{\sqrt{6} \cdot \sqrt{3} \cdot \sqrt{2}}{4 \cdot 3 \cdot 2}$$

Simplifying further, we get:

$$\frac{\sqrt{36}}{24}$$

Simplifying further, we get:

$$\frac{6}{24}$$

Simplifying further, we get:

$$\frac{1}{4}$$

Conclusion

In this article, we evaluated a complex expression involving trigonometric functions. We simplified the numerator and denominator separately and then divided the numerator by the denominator. We rationalized the denominator by multiplying the numerator and denominator by the conjugate of the denominator. Finally, we simplified the expression to get the final answer.

The final answer is $\boxed{\frac{1}{4}}$.

Introduction

In our previous article, we evaluated a complex expression involving trigonometric functions. We simplified the numerator and denominator separately and then divided the numerator by the denominator. We rationalized the denominator by multiplying the numerator and denominator by the conjugate of the denominator. Finally, we simplified the expression to get the final answer.

In this article, we will answer some frequently asked questions related to the evaluation of the expression and trigonometric functions.

Q&A

Q1: What is the value of $\operatorname{tg} 30^{\circ}$?

A1: The value of $\operatorname{tg} 30^{\circ}$ is $\frac{1}{\sqrt{3}}$.

Q2: What is the value of $\operatorname{sen} 45^{\circ}$?

A2: The value of $\operatorname{sen} 45^{\circ}$ is $\frac{1}{\sqrt{2}}$.

Q3: What is the value of $\csc 30^{\circ}$?

A3: The value of $\csc 30^{\circ}$ is $2$.

Q4: What is the value of $\sec 60^{\circ}$?

A4: The value of $\sec 60^{\circ}$ is $2$.

Q5: What is the value of $\operatorname{ctg} 45^{\circ}$?

A5: The value of $\operatorname{ctg} 45^{\circ}$ is $1$.

Q6: How do you simplify the expression $\frac{\sqrt{6}}{\sqrt{3} \cdot \sqrt{2}}$?

A6: To simplify the expression, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is $\sqrt{3} \cdot \sqrt{2}$.

Q7: How do you rationalize the denominator of the expression $\frac{\sqrt{6}}{4 \cdot \sqrt{3} \cdot \sqrt{2}}$?

A7: To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is $4 \cdot \sqrt{3} \cdot \sqrt{2}$.

Q8: What is the final answer to the expression $\frac{\sqrt{6} \cdot \sqrt{3} \cdot \sqrt{2}}{4 \cdot 3 \cdot 2}$?

A8: The final answer to the expression is $\frac{1}{4}$.

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the expression and trigonometric functions. We provided the values of various trigonometric functions and explained how to simplify and rationalize the denominator of the expression.

The final answer is $\boxed{\frac{1}{4}}$.

Additional Resources

For more information on trigonometric functions and their applications, please refer to the following resources:

• Trigonometry Wikipedia Page
• Trigonometric Functions Khan Academy Page
• Trigonometry Problems and Solutions

We hope this article has been helpful in understanding the evaluation of the expression and trigonometric functions. If you have any further questions or need additional clarification, please don't hesitate to ask.