Problem 48 - Contemporary Precalculus

Question

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{6 \tan t \sin t-3 \sin t}{9 \sin ^{2} t+3 \sin t}$$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(\frac{2\tan t - 1}{3\sin t + 1}\).

Step 1: Identify the GCD in the numerator and denominator

In the numerator, we have \(6\tan t\sin t - 3\sin t\). The GCD in this case is \(3\sin t\). In the denominator, we have \(9\sin^2 t + 3\sin t\). The GCD in this case is also \(3\sin t\).

Step 2: Factor out the GCD from the numerator and denominator

Now, factor out the GCD: Numerator: $$3\sin t(2\tan t - 1)$$ Denominator: $$3\sin t(3\sin t + 1)$$

Step 3: Cancel out common factors between the numerator and the denominator

We can see that both the numerator and denominator have a common factor of \(3\sin t\). We can now cancel out this common factor: $$\frac{3\sin t(2\tan t - 1)}{3\sin t(3\sin t + 1)} = \frac{2\tan t - 1}{3\sin t + 1}$$

Step 4: Write the final simplified expression

After canceling out the common factors, the simplified expression is: $$\frac{2\tan t - 1}{3\sin t + 1}$$

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics involves reducing them to their simplest form, often making them easier to work with. This can be achieved by performing operations such as combining like terms, canceling factors, or using identities to rewrite parts of the expression.
For instance, consider the trigonometric expression \( \frac{6 \tan t \sin t - 3 \sin t}{9 \sin^2 t + 3 \sin t} \). At first glance, it might seem complex, but by identifying and factoring out the Greatest Common Divisor (GCD), we can simplify it significantly.

• First, identify any common factors in the numerator and denominator.
• Factor these common elements out to rewrite the expression in a simpler form.
• Cancel the common elements, if possible, to further reduce the expression.

In our example, we were able to factor out \(3\sin t\), which simplified the expression to \( \frac{2\tan t - 1}{3\sin t + 1} \). Simplification helps us work more easily with equations and expressions, thereby making problem-solving more efficient.

GCD (Greatest Common Divisor)

The Greatest Common Divisor (GCD) is the largest number or expression that divides two or more numbers or expressions without leaving a remainder. In terms of algebraic expressions, it involves finding the common factors in the terms.
In the expression \( 6 \tan t \sin t - 3 \sin t \), both terms contain \(3\sin t\) as a factor. Thus, the GCD for the numerator is \(3\sin t\). Similarly, for the denominator \(9 \sin^2 t + 3 \sin t\), the GCD remains \(3\sin t\).

• Calculating the GCD involves identifying similar factors between terms.
• Once identified, these factors can be extracted, revealing a simpler form of the expression.
• This extraction aids significantly in simplifying complex expressions.

Understanding and applying the GCD in terms of algebraic expressions allow us to streamline complex calculations, laying the foundation for easier problem-solving techniques.

Factoring Expressions

Factoring expressions involves breaking down a polynomial or algebraic expression into its simplest components or factors that multiply together to form the original expression. Factoring is a crucial step in simplifying expressions and solving equations.
For the expression \(6\tan t \sin t - 3\sin t\), factoring involves pulling out the GCD, which is \(3\sin t\):
\(3\sin t(2\tan t - 1)\)

• Factoring helps in decomposing an expression into simpler terms.
• It allows for the cancellation of similar terms in fractions, hence simplifying the overall expression.
• This step-by-step breakdown is essential in changing the form of an algebraic equation, making further calculations or derivations more manageable.

Factoring is an indispensable algebraic tool that aids in both simplification and the solving of various mathematical problems. It’s a foundational technique necessary for dealing with equations and inequalities in calculus and algebra.