Simplify The Expression: $\left(\frac{2+\sin \theta}{3}\right)^3$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common expressions that require simplification is the one involving exponents and trigonometric functions. In this article, we will focus on simplifying the expression $\left(\frac{2+\sin \theta}{3}\right)^3$ using various mathematical techniques.

Understanding the Expression

The given expression is a cube of a fraction, where the numerator is a sum of a constant and a trigonometric function, and the denominator is a constant. To simplify this expression, we need to apply the rules of exponents and trigonometric identities.

Applying the Power Rule of Exponents

The power rule of exponents states that for any real number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$. We can apply this rule to the given expression by expanding the cube of the fraction.

$\left(\frac{2+\sin \theta}{3}\right)^3 = \frac{(2+\sin \theta)^3}{3^3}$

Expanding the Numerator

To expand the numerator, we can use the binomial theorem, which states that for any real number $a$ and $b$, and any positive integer $n$, $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

Applying the binomial theorem to the numerator, we get:

$(2+\sin \theta)^3 = \binom{3}{0} 2^3 + \binom{3}{1} 2^2 \sin \theta + \binom{3}{2} 2 \sin^2 \theta + \binom{3}{3} \sin^3 \theta$

Simplifying the Numerator

Simplifying the numerator, we get:

$(2+\sin \theta)^3 = 8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta$

Substituting the Simplified Numerator

Substituting the simplified numerator back into the original expression, we get:

$\left(\frac{2+\sin \theta}{3}\right)^3 = \frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{3^3}$

Simplifying the Expression

To simplify the expression further, we can divide the numerator and denominator by their greatest common divisor, which is 1.

$\left(\frac{2+\sin \theta}{3}\right)^3 = \frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{27}$

Conclusion

In this article, we simplified the expression $\left(\frac{2+\sin \theta}{3}\right)^3$ using various mathematical techniques, including the power rule of exponents and the binomial theorem. We expanded the numerator, simplified it, and substituted it back into the original expression. The final simplified expression is $\frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{27}$.

Final Answer

The final answer is $\boxed{\frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{27}}$.

Related Topics

• Simplifying expressions with exponents and trigonometric functions
• Applying the power rule of exponents
• Using the binomial theorem to expand expressions
• Simplifying expressions with trigonometric functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Trigonometry" by Charles P. McKeague

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we simplified the expression $\left(\frac{2+\sin \theta}{3}\right)^3$ using various mathematical techniques. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any real number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$. This rule allows us to simplify expressions with exponents by multiplying the exponents.

Q: How do I apply the power rule of exponents to the given expression?

A: To apply the power rule of exponents to the given expression, we need to expand the cube of the fraction. This can be done by multiplying the numerator and denominator by the cube root of the denominator.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are real numbers and $n$ is a positive integer.

Q: How do I use the binomial theorem to expand the numerator?

A: To use the binomial theorem to expand the numerator, we need to identify the values of $a$, $b$, and $n$. In this case, $a=2$, $b=\sin \theta$, and $n=3$. We can then apply the binomial theorem to expand the numerator.

Q: What is the simplified form of the numerator?

A: The simplified form of the numerator is $8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta$.

Q: How do I substitute the simplified numerator back into the original expression?

A: To substitute the simplified numerator back into the original expression, we need to replace the original numerator with the simplified form.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{27}$.

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

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

• Not applying the power rule of exponents correctly
• Not using the binomial theorem to expand expressions
• Not simplifying the numerator and denominator separately
• Not checking for common factors in the numerator and denominator

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\left(\frac{2+\sin \theta}{3}\right)^3$. We covered topics such as the power rule of exponents, the binomial theorem, and common mistakes to avoid when simplifying expressions.

Final Answer

The final answer is $\boxed{\frac{8 + 12 \sin \theta + 6 \sin^2 \theta + \sin^3 \theta}{27}}$.

Related Topics

• Simplifying expressions with exponents and trigonometric functions
• Applying the power rule of exponents
• Using the binomial theorem to expand expressions
• Simplifying expressions with trigonometric functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Trigonometry" by Charles P. McKeague

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.