Evaluate The Expression:A. $\tan ^{-1}(\sin (-3 \pi)$\] = ?

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Introduction

In mathematics, trigonometric functions and their inverses play a crucial role in solving various problems. The given expression, tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)), involves the inverse tangent function and the sine function. In this article, we will delve into the evaluation of this expression, exploring the properties of trigonometric functions and their inverses.

Understanding the Sine Function

The sine function, denoted by sin⁡x\sin x, is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function has a period of 2π2\pi, meaning that the value of the sine function repeats every 2π2\pi radians.

Evaluating sin⁡(−3π)\sin (-3 \pi)

To evaluate sin⁡(−3π)\sin (-3 \pi), we need to consider the periodic nature of the sine function. Since the sine function has a period of 2π2\pi, we can rewrite −3π-3 \pi as −3π+2π=−π-3 \pi + 2 \pi = -\pi. Therefore, sin⁡(−3π)=sin⁡(−π)\sin (-3 \pi) = \sin (-\pi).

Properties of the Sine Function

The sine function has several important properties that we need to consider when evaluating sin⁡(−π)\sin (-\pi). One of the key properties is that the sine function is an odd function, meaning that sin⁡(−x)=−sin⁡x\sin (-x) = -\sin x. Using this property, we can rewrite sin⁡(−π)\sin (-\pi) as −sin⁡π-\sin \pi.

Evaluating sin⁡π\sin \pi

The value of sin⁡π\sin \pi is 0, since the sine function is 0 at multiples of π\pi. Therefore, −sin⁡π=−0=0-\sin \pi = -0 = 0.

Understanding the Inverse Tangent Function

The inverse tangent function, denoted by tan⁡−1x\tan ^{-1} x, is a function that returns the angle whose tangent is equal to a given value. The range of the inverse tangent function is (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right).

Evaluating tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi))

Now that we have evaluated sin⁡(−3π)\sin (-3 \pi), we can substitute this value into the original expression. We have tan⁡−1(sin⁡(−3π))=tan⁡−1(0)\tan ^{-1}(\sin (-3 \pi)) = \tan ^{-1}(0).

Properties of the Inverse Tangent Function

The inverse tangent function has several important properties that we need to consider when evaluating tan⁡−1(0)\tan ^{-1}(0). One of the key properties is that the inverse tangent function is a continuous function, meaning that it has no gaps or discontinuities.

Evaluating tan⁡−1(0)\tan ^{-1}(0)

The value of tan⁡−1(0)\tan ^{-1}(0) is 0, since the tangent function is 0 at multiples of π\pi. Therefore, tan⁡−1(sin⁡(−3π))=tan⁡−1(0)=0\tan ^{-1}(\sin (-3 \pi)) = \tan ^{-1}(0) = 0.

Conclusion

In conclusion, the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) can be evaluated by considering the properties of the sine function and the inverse tangent function. By using the periodic nature of the sine function and the odd property of the sine function, we can rewrite sin⁡(−3π)\sin (-3 \pi) as 0. Then, by using the definition of the inverse tangent function, we can evaluate tan⁡−1(0)\tan ^{-1}(0) as 0. Therefore, the final answer to the expression is 0.

Final Answer

The final answer to the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) is 0.

Related Topics

  • Trigonometric functions and their inverses
  • Properties of the sine function
  • Properties of the inverse tangent function
  • Evaluating trigonometric expressions

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of evaluating the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)).

Introduction

In our previous article, we evaluated the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) and found that the final answer is 0. However, we understand that some readers may still have questions about the evaluation process. In this article, we will address some of the most frequently asked questions about evaluating the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)).

Q1: Why is the sine function periodic?

A1: The sine function is periodic because it represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. As the angle in the triangle changes, the ratio of the opposite side to the hypotenuse also changes, but it repeats itself after a certain interval, which is 2π2\pi radians.

Q2: What is the significance of the odd property of the sine function?

A2: The odd property of the sine function, sin⁡(−x)=−sin⁡x\sin (-x) = -\sin x, is important because it allows us to rewrite sin⁡(−3π)\sin (-3 \pi) as −sin⁡π-\sin \pi. This property is useful in evaluating trigonometric expressions involving negative angles.

Q3: Why is the inverse tangent function defined as tan⁡−1x\tan ^{-1} x?

A3: The inverse tangent function is defined as tan⁡−1x\tan ^{-1} x because it returns the angle whose tangent is equal to a given value. The range of the inverse tangent function is (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right), which means that the output of the inverse tangent function is always an angle between −π2-\frac{\pi}{2} and π2\frac{\pi}{2}.

Q4: How do we evaluate tan⁡−1(0)\tan ^{-1}(0)?

A4: To evaluate tan⁡−1(0)\tan ^{-1}(0), we need to consider the definition of the inverse tangent function. Since the tangent function is 0 at multiples of π\pi, we can conclude that tan⁡−1(0)=0\tan ^{-1}(0) = 0.

Q5: What is the final answer to the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi))?

A5: The final answer to the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) is 0, as we evaluated in our previous article.

Q6: Can we use the same method to evaluate other trigonometric expressions?

A6: Yes, we can use the same method to evaluate other trigonometric expressions involving the sine and inverse tangent functions. However, we need to consider the properties of the sine function and the inverse tangent function, as well as the specific values of the angles involved.

Q7: Are there any other ways to evaluate the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi))?

A7: Yes, there are other ways to evaluate the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)). One way is to use the identity tan⁡−1x=12ln⁡(1+x1−x)\tan ^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), which is valid for −1<x<1-1 < x < 1. However, this method requires more advanced calculus and is not necessary for this particular problem.

Q8: Can we use the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) in real-world applications?

A8: Yes, the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) can be used in real-world applications involving trigonometry and calculus. For example, it can be used to model the motion of a pendulum or the behavior of a physical system involving trigonometric functions.

Conclusion

In conclusion, the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) can be evaluated by considering the properties of the sine function and the inverse tangent function. By using the periodic nature of the sine function and the odd property of the sine function, we can rewrite sin⁡(−3π)\sin (-3 \pi) as 0. Then, by using the definition of the inverse tangent function, we can evaluate tan⁡−1(0)\tan ^{-1}(0) as 0. The final answer to the expression is 0.

Final Answer

The final answer to the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)) is 0.

Related Topics

  • Trigonometric functions and their inverses
  • Properties of the sine function
  • Properties of the inverse tangent function
  • Evaluating trigonometric expressions

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of evaluating the expression tan⁡−1(sin⁡(−3π))\tan ^{-1}(\sin (-3 \pi)).