Simplify $\frac{\sec(t) - \cos(t)}{\tan(t)}$ To A Single Trigonometric Function.

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Introduction

Trigonometric functions are an essential part of mathematics, and simplifying expressions involving these functions is a crucial skill for any math enthusiast. In this article, we will focus on simplifying the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)} to a single trigonometric function. We will use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Understanding the Expression

The given expression is sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)}. To simplify this expression, we need to understand the individual trigonometric functions involved. The secant function is the reciprocal of the cosine function, i.e., sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}. The tangent function is the ratio of the sine function to the cosine function, i.e., tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}.

Simplifying the Expression

To simplify the expression, we can start by rewriting the secant function in terms of the cosine function. We have sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}. Substituting this into the expression, we get:

1cos⁡(t)−cos⁡(t)tan⁡(t)\frac{\frac{1}{\cos(t)} - \cos(t)}{\tan(t)}

Using Trigonometric Identities

We can simplify the expression further by using the trigonometric identity tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}. Substituting this into the expression, we get:

1cos⁡(t)−cos⁡(t)sin⁡(t)cos⁡(t)\frac{\frac{1}{\cos(t)} - \cos(t)}{\frac{\sin(t)}{\cos(t)}}

Simplifying the Expression Further

We can simplify the expression further by multiplying the numerator and denominator by cos⁡(t)\cos(t). This gives us:

1−cos⁡2(t)sin⁡(t)\frac{1 - \cos^2(t)}{\sin(t)}

Using the Pythagorean Identity

We can simplify the expression further by using the Pythagorean identity sin⁡2(t)+cos⁡2(t)=1\sin^2(t) + \cos^2(t) = 1. Rearranging this identity, we get:

cos⁡2(t)=1−sin⁡2(t)\cos^2(t) = 1 - \sin^2(t)

Substituting this into the expression, we get:

1−(1−sin⁡2(t))sin⁡(t)\frac{1 - (1 - \sin^2(t))}{\sin(t)}

Simplifying the Expression to a Single Trigonometric Function

We can simplify the expression further by simplifying the numerator. We have:

1−(1−sin⁡2(t))sin⁡(t)=sin⁡2(t)sin⁡(t)\frac{1 - (1 - \sin^2(t))}{\sin(t)} = \frac{\sin^2(t)}{\sin(t)}

Final Result

We can simplify the expression further by canceling out the common factor of sin⁡(t)\sin(t). This gives us:

sin⁡2(t)sin⁡(t)=sin⁡(t)\frac{\sin^2(t)}{\sin(t)} = \sin(t)

Therefore, the simplified expression is sin⁡(t)\sin(t).

Conclusion

In this article, we simplified the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)} to a single trigonometric function. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The final result is sin⁡(t)\sin(t).

Frequently Asked Questions

Q: What is the secant function?

A: The secant function is the reciprocal of the cosine function, i.e., sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}.

Q: What is the tangent function?

A: The tangent function is the ratio of the sine function to the cosine function, i.e., tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}.

Q: How do you simplify the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)}?

A: To simplify the expression, you can start by rewriting the secant function in terms of the cosine function. You can then use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Q: What is the final result of simplifying the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)}?

A: The final result is sin⁡(t)\sin(t).

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Trigonometric Identities" by Math Open Reference

Additional Resources

  • [1] Khan Academy: Trigonometry
  • [2] MIT OpenCourseWare: Trigonometry
  • [3] Wolfram Alpha: Trigonometric Functions

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Introduction

In our previous article, we simplified the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)} to a single trigonometric function. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying trigonometric expressions.

Q&A

Q: What is the secant function?

A: The secant function is the reciprocal of the cosine function, i.e., sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}.

Q: What is the tangent function?

A: The tangent function is the ratio of the sine function to the cosine function, i.e., tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}.

Q: How do you simplify the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)}?

A: To simplify the expression, you can start by rewriting the secant function in terms of the cosine function. You can then use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Q: What is the final result of simplifying the expression sec⁡(t)−cos⁡(t)tan⁡(t)\frac{\sec(t) - \cos(t)}{\tan(t)}?

A: The final result is sin⁡(t)\sin(t).

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:

  • sin⁡2(t)+cos⁡2(t)=1\sin^2(t) + \cos^2(t) = 1
  • tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}
  • sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}
  • csc⁡(t)=1sin⁡(t)\csc(t) = \frac{1}{\sin(t)}

Q: How do you use the Pythagorean identity to simplify expressions?

A: The Pythagorean identity sin⁡2(t)+cos⁡2(t)=1\sin^2(t) + \cos^2(t) = 1 can be used to simplify expressions by rearranging it to cos⁡2(t)=1−sin⁡2(t)\cos^2(t) = 1 - \sin^2(t).

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

A: The sine function is the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function is the ratio of the adjacent side to the hypotenuse.

Q: How do you use the tangent function to simplify expressions?

A: The tangent function can be used to simplify expressions by using the identity tan⁡(t)=sin⁡(t)cos⁡(t)\tan(t) = \frac{\sin(t)}{\cos(t)}.

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

  • Not using the correct trigonometric identities
  • Not simplifying the expression fully
  • Not checking the final result for errors

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying trigonometric expressions. We covered common trigonometric identities, how to use the Pythagorean identity, and how to avoid common mistakes. By following these guidelines, you can simplify trigonometric expressions with confidence.

Frequently Asked Questions

Q: What is the best way to learn trigonometry?

A: The best way to learn trigonometry is to practice, practice, practice. Start with simple problems and gradually move on to more complex ones.

Q: What are some common applications of trigonometry?

A: Trigonometry has many applications in real-life situations, such as navigation, physics, engineering, and computer science.

Q: How do you use trigonometry in real-life situations?

A: Trigonometry is used in real-life situations such as calculating distances, heights, and angles.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Trigonometric Identities" by Math Open Reference

Additional Resources

  • [1] Khan Academy: Trigonometry
  • [2] MIT OpenCourseWare: Trigonometry
  • [3] Wolfram Alpha: Trigonometric Functions