Which Function Is Equivalent To Y = − Cot ⁡ ( X Y=-\cot (x Y = − Cot ( X ]?A. Y = − Tan ⁡ ( X Y=-\tan (x Y = − Tan ( X ]B. Y=-\tan \left(x+\frac{\pi}{2}\right ]C. Y = Tan ⁡ ( X Y=\tan (x Y = Tan ( X ]D. Y=\tan \left(x+\frac{\pi}{2}\right ]

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Introduction

Trigonometric functions are essential in mathematics, and understanding their properties and relationships is crucial for solving various mathematical problems. In this article, we will explore the equivalence of the function y=−cot⁡(x)y=-\cot (x) and its possible alternatives. We will examine each option and determine which one is equivalent to the given function.

Recalling the Definition of Cotangent

The cotangent function is defined as the reciprocal of the tangent function. Mathematically, it can be expressed as:

cot⁡(x)=1tan⁡(x)\cot (x) = \frac{1}{\tan (x)}

Analyzing the Options

Option A: y=−tan⁡(x)y=-\tan (x)

The tangent function is the reciprocal of the cotangent function. However, the negative sign in front of the tangent function does not change its reciprocal relationship with the cotangent function. Therefore, this option is not equivalent to the given function.

Option B: y=−tan⁡(x+π2)y=-\tan \left(x+\frac{\pi}{2}\right)

The tangent function has a period of π\pi, which means that it repeats every π\pi units. Adding π2\frac{\pi}{2} to the argument of the tangent function shifts its graph by π2\frac{\pi}{2} units. However, the negative sign in front of the tangent function still does not change its reciprocal relationship with the cotangent function. Therefore, this option is not equivalent to the given function.

Option C: y=tan⁡(x)y=\tan (x)

As mentioned earlier, the tangent function is the reciprocal of the cotangent function. However, the given function is y=−cot⁡(x)y=-\cot (x), which is the negative of the cotangent function. Therefore, this option is not equivalent to the given function.

Option D: y=tan⁡(x+π2)y=\tan \left(x+\frac{\pi}{2}\right)

This option is similar to Option B, but without the negative sign in front of the tangent function. The tangent function has a period of π\pi, which means that it repeats every π\pi units. Adding π2\frac{\pi}{2} to the argument of the tangent function shifts its graph by π2\frac{\pi}{2} units. However, the given function is y=−cot⁡(x)y=-\cot (x), which is the negative of the cotangent function. To make this option equivalent to the given function, we need to add a negative sign in front of the tangent function.

Conclusion

Based on the analysis of each option, we can conclude that the correct answer is:

Option D: y=tan⁡(x+π2)y=\tan \left(x+\frac{\pi}{2}\right)

However, to make this option equivalent to the given function, we need to add a negative sign in front of the tangent function. Therefore, the correct answer is:

y=−tan⁡(x+π2)y=-\tan \left(x+\frac{\pi}{2}\right)

Recap

In this article, we explored the equivalence of the function y=−cot⁡(x)y=-\cot (x) and its possible alternatives. We analyzed each option and determined which one is equivalent to the given function. We concluded that the correct answer is y=−tan⁡(x+π2)y=-\tan \left(x+\frac{\pi}{2}\right).

Key Takeaways

  • The cotangent function is the reciprocal of the tangent function.
  • The tangent function has a period of π\pi, which means that it repeats every π\pi units.
  • Adding π2\frac{\pi}{2} to the argument of the tangent function shifts its graph by π2\frac{\pi}{2} units.
  • The negative sign in front of the tangent function does not change its reciprocal relationship with the cotangent function.

Final Thoughts

Q: What is the difference between the tangent and cotangent functions?

A: The tangent function is the reciprocal of the cotangent function. Mathematically, it can be expressed as:

tan⁡(x)=1cot⁡(x)\tan (x) = \frac{1}{\cot (x)}

Q: What is the period of the tangent function?

A: The tangent function has a period of π\pi, which means that it repeats every π\pi units.

Q: How does the graph of the tangent function change when the argument is shifted by π2\frac{\pi}{2} units?

A: When the argument of the tangent function is shifted by π2\frac{\pi}{2} units, its graph is shifted by π2\frac{\pi}{2} units. This means that the new graph will be the same as the original graph, but shifted to the left or right by π2\frac{\pi}{2} units.

Q: What is the relationship between the tangent and cotangent functions?

A: The tangent function is the reciprocal of the cotangent function. Mathematically, it can be expressed as:

tan⁡(x)=1cot⁡(x)\tan (x) = \frac{1}{\cot (x)}

Q: How can we determine which option is equivalent to the given function y=−cot⁡(x)y=-\cot (x)?

A: To determine which option is equivalent to the given function y=−cot⁡(x)y=-\cot (x), we need to analyze each option and determine which one has the same properties and relationships as the given function.

Q: What is the correct answer to the question "Which function is equivalent to y=−cot⁡(x)y=-\cot (x)"?

A: The correct answer is y=−tan⁡(x+π2)y=-\tan \left(x+\frac{\pi}{2}\right).

Q: Why is the negative sign in front of the tangent function important?

A: The negative sign in front of the tangent function is important because it changes the sign of the function. In this case, the negative sign in front of the tangent function makes the function equivalent to the given function y=−cot⁡(x)y=-\cot (x).

Q: What is the key takeaway from this article?

A: The key takeaway from this article is that understanding the properties and relationships of trigonometric functions is essential for solving various mathematical problems.

Q: What is the final thought from this article?

A: The final thought from this article is that trigonometric functions are an essential part of mathematics, and understanding their properties and relationships is crucial for solving various mathematical problems.

Additional Resources

  • For more information on trigonometric functions, please refer to the following resources:
  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: Calculus and Linear Algebra
  • Wolfram MathWorld: Trigonometry

Conclusion

In this article, we have answered some frequently asked questions about trigonometric functions. We have discussed the difference between the tangent and cotangent functions, the period of the tangent function, and the relationship between the tangent and cotangent functions. We have also determined which option is equivalent to the given function y=−cot⁡(x)y=-\cot (x) and explained why the negative sign in front of the tangent function is important. We hope that this article has provided valuable insights and knowledge for readers who are interested in mathematics.