Simplifying Trigonometric Expressions: A Comprehensive Guide

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Understanding and simplifying trigonometric expressions is crucial for solving complex equations and problems. In this article, we will walk through the process of simplifying a specific trigonometric expression step by step, using fundamental trigonometric identities and algebraic manipulation techniques. This guide is designed to be accessible to students and professionals alike, making it easy to learn and apply these concepts.

Understanding the Expression

Our expression of interest is:

(frac{sin C cos C - 1.1 sin C}{sin C - cos C 1})

At first glance, this expression may appear complex, but by applying trigonometric identities and algebraic manipulation, we can simplify it significantly.

Simplifying the Expression

Let's break down the expression into simpler components and simplify it step-by-step:

1. The given expression is:

(frac{sin C cos C - 1.1 sin C}{sin C - cos C 1})

2. First, let's simplify the numerator:

Numerator: (sin C cos C - 1.1 sin C)

We can factor out (sin C) from the numerator:

(sin C (cos C - 1.1))

3. Now, let's simplify the denominator:

Denominator: (sin C - cos C 1)

The term (cos C 1) simplifies to (cos C), so the denominator becomes:

(sin C - cos C)

4. Combining the simplified numerator and denominator, we get:

(frac{sin C (cos C - 1.1)}{sin C - cos C})

5. Notice that the term (cos C - 1.1) in the numerator does not simplify further with the denominator, but the expression (sin C - cos C) is already in its simplest form.

Simplifying Further

Let's consider the expression again and simplify it further:

(frac{sin C cos C - 1.1 sin C}{sin C - cos C 1})

6. Rewriting the expression and simplifying the terms, we get:

(frac{sin C cos C - 1.1 sin C}{sin C - cos C})

7. Factoring out (sin C) from the numerator:

(frac{sin C (cos C - 1.1)}{sin C - cos C})

8. Simplifying the expression, we can cancel out (sin C) in the numerator and denominator:

(cos C - 1.1)

9. The expression simplifies to:

(cos C - 1.1)

Final Answer

Thus, the simplified form of the given expression is:

(cos C - 1.1)

Conclusion

The process of simplifying the given trigonometric expression highlights the importance of fundamental trigonometric identities and algebraic manipulation. Understanding these techniques allows us to break down complex expressions into simpler, more manageable parts, making it easier to solve problems involving trigonometric functions.

Key Takeaways

Trigonometric identities are essential tools for simplifying expressions. Algebraic manipulation can help streamline the simplification process. The expression simplifies to (cos C - 1.1).

Related Keywords

Simplifying Trigonometric Expressions, Trigonometric Identities, Algebraic Manipulation, Trigonometric Functions, Math Problems