Evaluate The Expression: ${ 3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} =\$} A. { -\sqrt{3}$}$B. 0C. { \sqrt{3}$}$D. ${$2 \sqrt{3}$}$85. Which Of The Following Is Correct About An Angle

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Introduction

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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 evaluating trigonometric expressions, specifically the given expression: $3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ}$. We will break down the solution step by step and provide a comprehensive explanation of the trigonometric concepts involved.

Understanding the Given Expression

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The given expression is a combination of three trigonometric functions: tangent, sine, and cosine. The expression is: $3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ}$. To evaluate this expression, we need to understand the properties of each trigonometric function and how they behave with respect to the angle.

Tangent Function

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The tangent function is defined as the ratio of the sine and cosine functions: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. The tangent function has a periodicity of $180^{\circ}$, which means that the value of the tangent function repeats every $180^{\circ}$.

Sine Function

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The sine function is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. The sine function has a periodicity of $360^{\circ}$, which means that the value of the sine function repeats every $360^{\circ}$.

Cosine Function

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The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function has a periodicity of $360^{\circ}$, which means that the value of the cosine function repeats every $360^{\circ}$.

Evaluating the Expression

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To evaluate the given expression, we need to simplify each trigonometric function separately and then combine the results.

Evaluating the Tangent Function

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The tangent function is evaluated at $930^{\circ}$. Since the tangent function has a periodicity of $180^{\circ}$, we can rewrite $930^{\circ}$ as $930^{\circ} - 6 \cdot 180^{\circ} = 30^{\circ}$. Therefore, $\tan 930^{\circ} = \tan 30^{\circ} = \frac{1}{\sqrt{3}}$.

Evaluating the Sine Function

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The sine function is evaluated at $1200^{\circ}$. Since the sine function has a periodicity of $360^{\circ}$, we can rewrite $1200^{\circ}$ as $1200^{\circ} - 3 \cdot 360^{\circ} = 120^{\circ}$. Therefore, $\sin 1200^{\circ} = \sin 120^{\circ} = \frac{\sqrt{3}}{2}$.

Evaluating the Cosine Function

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The cosine function is evaluated at $1410^{\circ}$. Since the cosine function has a periodicity of $360^{\circ}$, we can rewrite $1410^{\circ}$ as $1410^{\circ} - 3 \cdot 360^{\circ} = 150^{\circ}$. Therefore, $\cos 1410^{\circ} = \cos 150^{\circ} = -\frac{\sqrt{3}}{2}$.

Combining the Results

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Now that we have evaluated each trigonometric function separately, we can combine the results to evaluate the given expression.

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = 3 \cdot \frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2} - \left(-\frac{\sqrt{3}}{2}\right)$

Simplifying the expression, we get:

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = \frac{3}{\sqrt{3}} + \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}$

Combining the like terms, we get:

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = \frac{3}{\sqrt{3}} + \sqrt{3}$

Rationalizing the denominator, we get:

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = \frac{3\sqrt{3}}{3} + \sqrt{3}$

Simplifying the expression, we get:

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = \sqrt{3} + \sqrt{3}$

Combining the like terms, we get:

$3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ} = 2\sqrt{3}$

Therefore, the value of the given expression is $2\sqrt{3}$.

Conclusion

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In this article, we evaluated the given expression: $3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ}$. We broke down the solution step by step and provided a comprehensive explanation of the trigonometric concepts involved. We simplified each trigonometric function separately and then combined the results to evaluate the given expression. The final answer is $2\sqrt{3}$.

Frequently Asked Questions

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Q: What is the periodicity of the tangent function?

A: The periodicity of the tangent function is $180^{\circ}$.

Q: What is the periodicity of the sine function?

A: The periodicity of the sine function is $360^{\circ}$.

Q: What is the periodicity of the cosine function?

A: The periodicity of the cosine function is $360^{\circ}$.

Q: How do you evaluate the tangent function at a given angle?

A: To evaluate the tangent function at a given angle, you need to rewrite the angle in terms of the periodicity of the tangent function.

Q: How do you evaluate the sine function at a given angle?

A: To evaluate the sine function at a given angle, you need to rewrite the angle in terms of the periodicity of the sine function.

Q: How do you evaluate the cosine function at a given angle?

A: To evaluate the cosine function at a given angle, you need to rewrite the angle in terms of the periodicity of the cosine function.

References

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• [1] "Trigonometry" by Michael Corral
• [2] "Trigonometry" by I. M. Gelfand
• [3] "Trigonometry" by Charles P. McKeague

Note: The references provided are for general information purposes only and are not specific to the given expression.

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Q&A: Evaluating Trigonometric Expressions

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Q: What is the value of the expression $3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ}$?

A: The value of the expression $3 \tan 930^{\circ} + \sin 1200^{\circ} - \cos 1410^{\circ}$ is $2\sqrt{3}$.

Q: How do you evaluate the tangent function at a given angle?

A: To evaluate the tangent function at a given angle, you need to rewrite the angle in terms of the periodicity of the tangent function. The periodicity of the tangent function is $180^{\circ}$.

Q: How do you evaluate the sine function at a given angle?

A: To evaluate the sine function at a given angle, you need to rewrite the angle in terms of the periodicity of the sine function. The periodicity of the sine function is $360^{\circ}$.

Q: How do you evaluate the cosine function at a given angle?

A: To evaluate the cosine function at a given angle, you need to rewrite the angle in terms of the periodicity of the cosine function. The periodicity of the cosine function is $360^{\circ}$.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that are used to describe the relationships between the sides and angles of triangles. However, the sine function is defined as the ratio of the opposite side to the hypotenuse, while the cosine function is defined as the ratio of the adjacent side to the hypotenuse.

Q: What is the value of the expression $\tan 45^{\circ}$?

A: The value of the expression $\tan 45^{\circ}$ is 1.

Q: What is the value of the expression $\sin 90^{\circ}$?

A: The value of the expression $\sin 90^{\circ}$ is 1.

Q: What is the value of the expression $\cos 0^{\circ}$?

A: The value of the expression $\cos 0^{\circ}$ is 1.

Q: How do you simplify a trigonometric expression?

A: To simplify a trigonometric expression, you need to use the properties of the trigonometric functions, such as the sum and difference formulas, and the Pythagorean identity.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental property of the trigonometric functions that states that $\sin^2 \theta + \cos^2 \theta = 1$.

Q: How do you use the Pythagorean identity to simplify a trigonometric expression?

A: To use the Pythagorean identity to simplify a trigonometric expression, you need to rewrite the expression in terms of the sine and cosine functions, and then use the Pythagorean identity to simplify the expression.

Common Trigonometric Identities

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Pythagorean Identity

$\sin^2 \theta + \cos^2 \theta = 1$

Sum Formula for Sine

$\sin (A + B) = \sin A \cos B + \cos A \sin B$

Sum Formula for Cosine

$\cos (A + B) = \cos A \cos B - \sin A \sin B$

Difference Formula for Sine

$\sin (A - B) = \sin A \cos B - \cos A \sin B$

Difference Formula for Cosine

$\cos (A - B) = \cos A \cos B + \sin A \sin B$

Trigonometric Functions of Common Angles

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Trigonometric Functions of 0°

$\sin 0^{\circ} = 0$ $\cos 0^{\circ} = 1$ $\tan 0^{\circ} = 0$

Trigonometric Functions of 30°

$\sin 30^{\circ} = \frac{1}{2}$ $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$

Trigonometric Functions of 45°

$\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$ $\tan 45^{\circ} = 1$

Trigonometric Functions of 60°

$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ $\cos 60^{\circ} = \frac{1}{2}$ $\tan 60^{\circ} = \sqrt{3}$

Trigonometric Functions of 90°

$\sin 90^{\circ} = 1$ $\cos 90^{\circ} = 0$ $\tan 90^{\circ} = \infty$

Conclusion

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In this article, we have provided a comprehensive guide to evaluating trigonometric expressions. We have covered the basics of trigonometry, including the definition of the trigonometric functions, the properties of the trigonometric functions, and the common trigonometric identities. We have also provided a list of common trigonometric functions of common angles. We hope that this article has been helpful in providing a clear understanding of the subject.