Simplify The Following Expression: $ \frac{\sin 3x + \sin 7x}{\cos 3x + \cos 7x} }$[HINT ${$3x = (5x - 2x)$ $ And ${ 7x = (5x + 2x)\$} ]

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Introduction

Trigonometric expressions can be simplified using various identities and formulas. In this article, we will simplify the given expression using the sum-to-product identities for sine and cosine functions. The given expression is:

sin⁡3x+sin⁡7xcos⁡3x+cos⁡7x\frac{\sin 3x + \sin 7x}{\cos 3x + \cos 7x}

Step 1: Apply the Sum-to-Product Identity for Sine

The sum-to-product identity for sine is given by:

sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

We can apply this identity to the numerator of the given expression:

sin⁡3x+sin⁡7x=2sin⁡(3x+7x2)cos⁡(3x−7x2)\sin 3x + \sin 7x = 2\sin \left(\frac{3x+7x}{2}\right)\cos \left(\frac{3x-7x}{2}\right)

=2sin⁡5xcos⁡(−2x)= 2\sin 5x\cos (-2x)

Since cos⁡(−θ)=cos⁡θ\cos (-\theta) = \cos \theta, we have:

=2sin⁡5xcos⁡2x= 2\sin 5x\cos 2x

Step 2: Apply the Sum-to-Product Identity for Cosine

The sum-to-product identity for cosine is given by:

cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)\cos A + \cos B = 2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

We can apply this identity to the denominator of the given expression:

cos⁡3x+cos⁡7x=2cos⁡(3x+7x2)cos⁡(3x−7x2)\cos 3x + \cos 7x = 2\cos \left(\frac{3x+7x}{2}\right)\cos \left(\frac{3x-7x}{2}\right)

=2cos⁡5xcos⁡(−2x)= 2\cos 5x\cos (-2x)

Since cos⁡(−θ)=cos⁡θ\cos (-\theta) = \cos \theta, we have:

=2cos⁡5xcos⁡2x= 2\cos 5x\cos 2x

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can rewrite the original expression as:

2sin⁡5xcos⁡2x2cos⁡5xcos⁡2x\frac{2\sin 5x\cos 2x}{2\cos 5x\cos 2x}

We can cancel out the common factors of 2cos⁡2x2\cos 2x:

=sin⁡5xcos⁡5x= \frac{\sin 5x}{\cos 5x}

Step 4: Apply the Reciprocal Identity

The reciprocal identity for tangent is given by:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

We can rewrite the simplified expression as:

=tan⁡5x= \tan 5x

Conclusion

In this article, we simplified the given trigonometric expression using the sum-to-product identities for sine and cosine functions. We applied the identities to the numerator and denominator, canceled out common factors, and finally applied the reciprocal identity to obtain the final result.

Final Answer

The final answer is tan⁡5x\boxed{\tan 5x}.

Discussion

This problem is a great example of how to simplify trigonometric expressions using various identities and formulas. The sum-to-product identities are particularly useful when dealing with expressions that involve the sum or difference of two angles.

Related Problems

  • Simplify the expression sin⁡2x+sin⁡4xcos⁡2x+cos⁡4x\frac{\sin 2x + \sin 4x}{\cos 2x + \cos 4x} using the sum-to-product identities.
  • Simplify the expression cos⁡3x+cos⁡5xsin⁡3x+sin⁡5x\frac{\cos 3x + \cos 5x}{\sin 3x + \sin 5x} using the sum-to-product identities.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak

Tags

  • Trigonometry
  • Sum-to-product identities
  • Reciprocal identity
  • Simplification of trigonometric expressions

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Introduction

In our previous article, we simplified the given trigonometric expression using the sum-to-product identities for sine and cosine functions. In this article, we will answer some frequently asked questions related to simplifying trigonometric expressions.

Q1: What are the sum-to-product identities for sine and cosine?

The sum-to-product identities for sine and cosine are given by:

sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)\cos A + \cos B = 2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

Q2: How do I apply the sum-to-product identities to simplify a trigonometric expression?

To apply the sum-to-product identities, you need to identify the angles in the expression and then use the identities to rewrite the expression in a simpler form.

Q3: What is the reciprocal identity for tangent?

The reciprocal identity for tangent is given by:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Q4: How do I simplify a trigonometric expression using the reciprocal identity?

To simplify a trigonometric expression using the reciprocal identity, you need to rewrite the expression in the form sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta} and then use the identity to simplify it.

Q5: What are some common mistakes to avoid when simplifying trigonometric expressions?

Some common mistakes to avoid when simplifying trigonometric expressions include:

  • Not using the correct identities
  • Not canceling out common factors
  • Not rewriting the expression in the correct form

Q6: How do I know which identity to use when simplifying a trigonometric expression?

To determine which identity to use, you need to examine the expression and identify the angles and trigonometric functions involved. Then, you can choose the appropriate identity to simplify the expression.

Q7: Can I use the sum-to-product identities to simplify expressions involving the difference of two angles?

Yes, you can use the sum-to-product identities to simplify expressions involving the difference of two angles. However, you need to rewrite the expression in the form sin⁡(A−B)\sin (A-B) or cos⁡(A−B)\cos (A-B) before applying the identity.

Q8: How do I simplify expressions involving the product of two trigonometric functions?

To simplify expressions involving the product of two trigonometric functions, you can use the product-to-sum identities. These identities are given by:

sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin (A+B) + \sin (A-B)]

cos⁡Acos⁡B=12[cos⁡(A+B)+cos⁡(A−B)]\cos A \cos B = \frac{1}{2}[\cos (A+B) + \cos (A-B)]

Q9: Can I use the reciprocal identity to simplify expressions involving the quotient of two trigonometric functions?

Yes, you can use the reciprocal identity to simplify expressions involving the quotient of two trigonometric functions. However, you need to rewrite the expression in the form sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta} before applying the identity.

Q10: How do I check my work when simplifying trigonometric expressions?

To check your work, you can use the following steps:

  • Plug in some values for the angles and trigonometric functions involved
  • Simplify the expression using the identities
  • Check that the simplified expression is equivalent to the original expression

Final Answer

The final answer is tan⁡5x\boxed{\tan 5x}.

Discussion

Simplifying trigonometric expressions is an important skill in mathematics and physics. By using the sum-to-product identities, reciprocal identity, and product-to-sum identities, you can simplify complex expressions and solve problems more efficiently.

Related Problems

  • Simplify the expression sin⁡2x+sin⁡4xcos⁡2x+cos⁡4x\frac{\sin 2x + \sin 4x}{\cos 2x + \cos 4x} using the sum-to-product identities.
  • Simplify the expression cos⁡3x+cos⁡5xsin⁡3x+sin⁡5x\frac{\cos 3x + \cos 5x}{\sin 3x + \sin 5x} using the sum-to-product identities.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak

Tags

  • Trigonometry
  • Sum-to-product identities
  • Reciprocal identity
  • Product-to-sum identities
  • Simplification of trigonometric expressions