$\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A}$ Simplifies To What?

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Introduction

Trigonometric expressions are a fundamental part of mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the simplification of a specific trigonometric expression: sin⁑3Asin⁑Aβˆ’cos⁑3Acos⁑A\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A}. We will use various trigonometric identities and formulas to simplify this expression and arrive at a final result.

Background

Before we dive into the simplification process, let's review some basic trigonometric identities and formulas. The sine and cosine functions are periodic with a period of 2Ο€2\pi, and they can be expressed in terms of each other using the following identities:

  • sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1
  • sin⁑(A+B)=sin⁑Acos⁑B+cos⁑Asin⁑B\sin (A+B) = \sin A \cos B + \cos A \sin B
  • cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos (A+B) = \cos A \cos B - \sin A \sin B

We will use these identities to simplify the given expression.

Simplifying the Expression

To simplify the expression sin⁑3Asin⁑Aβˆ’cos⁑3Acos⁑A\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A}, we can start by using the sum and difference formulas for sine and cosine:

  • sin⁑3A=sin⁑(2A+A)=sin⁑2Acos⁑A+cos⁑2Asin⁑A\sin 3A = \sin (2A+A) = \sin 2A \cos A + \cos 2A \sin A
  • cos⁑3A=cos⁑(2A+A)=cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑A\cos 3A = \cos (2A+A) = \cos 2A \cos A - \sin 2A \sin A

Substituting these expressions into the original expression, we get:

sin⁑3Asin⁑Aβˆ’cos⁑3Acos⁑A=sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑A\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} = \frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A}

Using Trigonometric Identities

Now, let's use the trigonometric identities to simplify the expression further. We can start by using the identity sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1 to rewrite the expression:

sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑A=sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A\frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A} = \frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A}

Simplifying the expression, we get:

sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A=sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A\frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A} = \frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A}

Using the Double Angle Formulas

Now, let's use the double angle formulas to simplify the expression further. We can start by using the formula sin⁑2A=2sin⁑Acos⁑A\sin 2A = 2\sin A \cos A to rewrite the expression:

sin⁑2Acos⁑A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’sin⁑2Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A=2sin⁑Acos⁑2A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A\frac{\sin 2A \cos A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - \sin 2A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A} = \frac{2\sin A \cos^2 A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A}

Simplifying the expression, we get:

2sin⁑Acos⁑2A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A=2cos⁑2A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A\frac{2\sin A \cos^2 A + \cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A} = 2\cos^2 A + \frac{\cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A}

Using the Half Angle Formulas

Now, let's use the half angle formulas to simplify the expression further. We can start by using the formula cos⁑2A=2cos⁑2Aβˆ’1\cos 2A = 2\cos^2 A - 1 to rewrite the expression:

2cos⁑2A+cos⁑2Asin⁑Asin⁑Aβˆ’cos⁑2Acos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A=2cos⁑2A+(2cos⁑2Aβˆ’1)sin⁑Asin⁑Aβˆ’(2cos⁑2Aβˆ’1)cos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A2\cos^2 A + \frac{\cos 2A \sin A}{\sin A} - \frac{\cos 2A \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A} = 2\cos^2 A + \frac{(2\cos^2 A - 1) \sin A}{\sin A} - \frac{(2\cos^2 A - 1) \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A}

Simplifying the expression, we get:

2cos⁑2A+(2cos⁑2Aβˆ’1)sin⁑Asin⁑Aβˆ’(2cos⁑2Aβˆ’1)cos⁑Aβˆ’2sin⁑Acos⁑Asin⁑Acos⁑Aβ‹…sin⁑Asin⁑A=2cos⁑2A+2cos⁑2Aβˆ’1βˆ’(2cos⁑2Aβˆ’1)βˆ’2sin⁑Acos⁑Asin⁑A2\cos^2 A + \frac{(2\cos^2 A - 1) \sin A}{\sin A} - \frac{(2\cos^2 A - 1) \cos A - 2\sin A \cos A \sin A}{\cos A} \cdot \frac{\sin A}{\sin A} = 2\cos^2 A + 2\cos^2 A - 1 - (2\cos^2 A - 1) - 2\sin A \cos A \sin A

Final Simplification

Now, let's simplify the expression further:

2cos⁑2A+2cos⁑2Aβˆ’1βˆ’(2cos⁑2Aβˆ’1)βˆ’2sin⁑Acos⁑Asin⁑A=4cos⁑2Aβˆ’2βˆ’2sin⁑2Acos⁑2A2\cos^2 A + 2\cos^2 A - 1 - (2\cos^2 A - 1) - 2\sin A \cos A \sin A = 4\cos^2 A - 2 - 2\sin^2 A \cos^2 A

Using the identity sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1, we can rewrite the expression as:

4cos⁑2Aβˆ’2βˆ’2sin⁑2Acos⁑2A=4cos⁑2Aβˆ’2βˆ’2(1βˆ’cos⁑2A)cos⁑2A4\cos^2 A - 2 - 2\sin^2 A \cos^2 A = 4\cos^2 A - 2 - 2(1 - \cos^2 A) \cos^2 A

Simplifying the expression, we get:

4cos⁑2Aβˆ’2βˆ’2(1βˆ’cos⁑2A)cos⁑2A=4cos⁑2Aβˆ’2βˆ’2cos⁑2A+2cos⁑4A4\cos^2 A - 2 - 2(1 - \cos^2 A) \cos^2 A = 4\cos^2 A - 2 - 2\cos^2 A + 2\cos^4 A

Final Result

Now, let's simplify the expression further:

4cos⁑2Aβˆ’2βˆ’2cos⁑2A+2cos⁑4A=2cos⁑2Aβˆ’2+2cos⁑4A4\cos^2 A - 2 - 2\cos^2 A + 2\cos^4 A = 2\cos^2 A - 2 + 2\cos^4 A

Factoring out the common term 2cos⁑2A2\cos^2 A, we get:

2cos⁑2Aβˆ’2+2cos⁑4A=2cos⁑2A(1βˆ’cos⁑2A)βˆ’22\cos^2 A - 2 + 2\cos^4 A = 2\cos^2 A(1 - \cos^2 A) - 2

Using the identity sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1, we can rewrite the expression as:

2cos⁑2A(1βˆ’cos⁑2A)βˆ’2=2cos⁑2Asin⁑2Aβˆ’22\cos^2 A(1 - \cos^2 A) - 2 = 2\cos^2 A \sin^2 A - 2

Conclusion

In this article, we simplified the trigonometric expression sin⁑3Asin⁑Aβˆ’cos⁑3Acos⁑A\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} using various trigonometric identities and formulas. We started by using the sum and difference formulas for sine and cosine, and then used the double angle and half angle formulas to simplify the expression further. Finally, we arrived at the final result: 2cos⁑2Asin⁑2Aβˆ’22\cos^2 A \sin^2 A - 2. This result can be used to solve various trigonometric problems and equations.

Introduction

In our previous article, we simplified the trigonometric expression sin⁑3Asin⁑Aβˆ’cos⁑3Acos⁑A\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} using various trigonometric identities and formulas. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of trigonometric expressions.

Q: What are some common trigonometric identities used in simplifying expressions?

A: Some common trigonometric identities used in simplifying expressions include:

  • sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1
  • sin⁑(A+B)=sin⁑Acos⁑B+cos⁑Asin⁑B\sin (A+B) = \sin A \cos B + \cos A \sin B
  • cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos (A+B) = \cos A \cos B - \sin A \sin B
  • sin⁑2A=2sin⁑Acos⁑A\sin 2A = 2\sin A \cos A
  • cos⁑2A=2cos⁑2Aβˆ’1\cos 2A = 2\cos^2 A - 1

Q: How do I use the sum and difference formulas for sine and cosine?

A: To use the sum and difference formulas for sine and cosine, you need to identify the angles involved in the expression. Then, you can apply the formulas to simplify the expression. For example, if you have the expression sin⁑(A+B)\sin (A+B), you can use the sum formula for sine to rewrite it as sin⁑Acos⁑B+cos⁑Asin⁑B\sin A \cos B + \cos A \sin B.

Q: What is the difference between the double angle and half angle formulas?

A: The double angle formulas are used to express a trigonometric function in terms of the same function with a double angle. For example, the double angle formula for sine is sin⁑2A=2sin⁑Acos⁑A\sin 2A = 2\sin A \cos A. The half angle formulas, on the other hand, are used to express a trigonometric function in terms of the same function with a half angle. For example, the half angle formula for cosine is cos⁑A2=±1+cos⁑A2\cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}.

Q: How do I use the trigonometric identities to simplify expressions?

A: To use the trigonometric identities to simplify expressions, you need to identify the identities that can be applied to the expression. Then, you can apply the identities to simplify the expression. For example, if you have the expression sin⁑2A+cos⁑2A\sin^2 A + \cos^2 A, you can use the identity sin⁑2A+cos⁑2A=1\sin^2 A + \cos^2 A = 1 to rewrite it as 11.

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
  • Not using the correct order of operations

Q: How do I check my work when simplifying trigonometric expressions?

A: To check your work when simplifying trigonometric expressions, you need to:

  • Verify that you have used the correct trigonometric identities
  • Verify that you have simplified the expression fully
  • Verify that the final result is correct
  • Use a calculator or computer software to check the final result

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the simplification of trigonometric expressions. We covered topics such as common trigonometric identities, sum and difference formulas, double angle and half angle formulas, and common mistakes to avoid. We also provided tips on how to check your work when simplifying trigonometric expressions. By following these tips and using the correct trigonometric identities, you can simplify trigonometric expressions with confidence.

Additional Resources

For more information on simplifying trigonometric expressions, you can refer to the following resources:

  • "Trigonometry" by Michael Corral
  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague and Mark D. Turner
  • Online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.

Final Thoughts

Simplifying trigonometric expressions is an essential skill for students and professionals in mathematics and science. By mastering the trigonometric identities and formulas, you can simplify complex expressions and solve problems with ease. Remember to always check your work and use the correct order of operations to ensure that your final result is correct. With practice and patience, you can become proficient in simplifying trigonometric expressions and tackle even the most challenging problems.