Problem 10 - Precalculus

Question

Carry out the indicated operations. (a) \(\frac{5-2 T}{2 T-5}\) (b) \(\frac{5-2 \tan \theta}{2 \tan \theta-5}\)

Step-by-Step Solution

Answer

Both expressions simplify to (a) -1 (b) -1.

Step 1: Identify the Expression

The given expressions are (a) \( \frac{5-2T}{2T-5} \) and (b) \( \frac{5-2\tan \theta}{2\tan \theta-5} \). Note how similar these two expressions look with slight differences in variables.

Step 2: Simplify the Expressions

Both expressions can be related to each other by recognizing that \( 2T - 5 \) is the negative of \( 5 - 2T \). Therefore, any expression of the form \( \frac{a-b}{b-a} \) simplifies to \( -1 \).

Step 3: Apply Simplification to Expression (a)

For expression (a), \( 5 - 2T \) and \( 2T - 5 \) are opposites, so we can say \( \frac{5-2T}{2T-5} = -1 \).

Step 4: Apply Simplification to Expression (b)

For expression (b), \( 5 - 2\tan \theta \) and \( 2\tan \theta - 5 \) are opposites. So, we similarly conclude that \( \frac{5-2\tan \theta}{2\tan \theta-5} = -1 \).

Key Concepts

Operations on Fractions

Working with fractions is a critical skill in algebra. We often encounter complex fractions, which may seem complicated at first. Understanding that fractions involve a numerator and a denominator is fundamental. When we perform operations like addition, subtraction, or simplification, always keep this in mind.

Here’s a simple way to approach working with fractions:

• Ensure both fractions have a common denominator when adding or subtracting.
• Multiply numerators and denominators straight across when multiplying fractions.
• Flip the second fraction and multiply when dividing fractions.

In the given exercise, we noted that both fractions were in the form \( \frac{a-b}{b-a} \), which simplifies to \( -1 \). Recognizing such patterns can significantly simplify calculations.

Algebraic expressions may involve variables instead of just numbers, as in \( \frac{5-2T}{2T-5} \). The concept is the same: focus on understanding the relationship between the parts of your expression to simplify it effectively.

Trigonometric Identities

Trigonometric functions like sine, cosine, and tangent are used to relate the angles of a triangle to the lengths of its sides. Among these, tangent (\(\tan\)) is especially useful in calculations involving angles.

• Tangent of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
• There are key trigonometric identities that express relationships among these functions.

In our exercise, \( \tan \theta \) appears in a fraction. The fraction \( \frac{5 - 2 \tan \theta}{2 \tan \theta - 5} \) reduces to \( -1 \).

When simplifying expressions involving trigonometric functions, always look for these identities and patterns. They often lead to significant simplification.

Expression Simplification

Simplifying expressions is all about eliminating unnecessary complexity. When you simplify, you want to rewrite expressions in their simplest forms. This is crucial for both understanding and solving algebraic problems.

In the given exercise, recognizing the structure \( \frac{a - b}{b - a} = -1 \) is an example of expression simplification. Here's why:

• \( a-b \) and \( b-a \) are negatives of each other. Their division results in \( -1 \).
• Simplification helps to reveal these underlying relationships.
• Often, the simplified form of an expression conveys more information subtly and concisely.

When you simplify algebraic expressions, keep your focus on combining like terms, reducing fractions, and recognizing patterns such as identities or symmetries. This allows for balanced equations and reveals critical results which can be used to solve more complex problems.

The ability to simplify expressions forms a cornerstone in mastering algebra and paves the way for tackling more advanced topics.