Problem 11 - Precalculus with Calculus Previews

Question

Reduce the given expression to a single trigonometric function. $$ \frac{\sin t+\sin t \cos t}{1+\cos t} $$

Step-by-Step Solution

Answer

The given expression simplifies to \( \sin t \).

Step 1: Factor Out Common Terms

First, notice that both terms in the numerator have a common factor \( \sin t \). Factor \( \sin t \) out from the numerator to simplify the expression:\[ \frac{\sin t + \sin t \cos t}{1 + \cos t} = \frac{\sin t (1 + \cos t)}{1 + \cos t}\]

Step 2: Cancel Common Factors

Now that you've factored the numerator, you can cancel the common factor \( 1 + \cos t \) from both the numerator and the denominator:\[ \frac{\sin t (1 + \cos t)}{1 + \cos t} = \sin t\]

Step 3: Simplified Expression

After cancelling the common terms, the simplified expression is \( \sin t \). This is now a single trigonometric function.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex mathematical expressions easier to handle. It's like tidying up a messy room, finding ways to combine or reduce terms to their simplest form. Trigonometric identities, such as \(\sin t\), \(\cos t\), and others, are powerful tools for simplifying expressions. Here, we're tasked with reducing a fraction to just one trigonometric function.

• Look for common terms that can be combined or factored out.
• Utilize identities or known simplifications to reduce the complexity of expressions.

For example, in the given expression, \(\frac{\sin t + \sin t \cos t}{1 + \cos t}\), the goal was to simplify it to \(\sin t\). By simplifying, we reduce calculation errors and make further steps more manageable. Breaking down expressions into simpler parts can be very helpful in solving calculus problems or other equations.

Factoring

Factoring is an essential technique in mathematics used to break down expressions into simpler, multiply connected parts. Think of it as taking a number or equation and expressing it as a product of its factors, similar to taking apart a puzzle to see its pieces. For trigonometric identities, recognizing and factoring out common terms can greatly simplify the process. In the exercise you're dealing with, both terms in the numerator have a common factor, \(\sin t\) - which can be factored out:

• Express the expression as a product of simplified terms.
• Use common factors to reveal simpler underlying relationships that might not be initially obvious.

By factoring \(\sin t\) out from \(\sin t + \sin t \cos t\), you're left with \(\sin t(1 + \cos t)\). This not only simplifies calculations but also sets up the opportunity to remove further redundancies in the expression.

Cancelling Common Factors

Canceling common factors is a straightforward but powerful step in simplifying mathematical expressions. It involves reducing an expression by dividing both the numerator and the denominator by the same non-zero factor. It’s like cleaning up the last bits of clutter after arranging the room - you look for final components that appear in both the top and bottom part of a fraction.In the trigonometric expression \(\frac{\sin t (1 + \cos t)}{1 + \cos t}\), you can see \(1 + \cos t\) is present in both the numerator and denominator. Since they are non-zero, this common factor can be cancelled out, simplifying the expression further to just \(\sin t\).

• Identify and cancel out repeating terms in numerical fractions or algebraic expressions.
• Ensure that the factors being cancelled are the same in both parts of the expression.

This step is crucial as it reduces the expression to its simplest form, eliminating unnecessary complexity and yielding the final result effectively.