Simplify the trigonometric expression.
4 + 7cos(y)/
7 + 4sec(y)


To simplify the trigonometric expression:

\[
\frac{4 + 7 \cos(y)}{7 + 4 \sec(y)}
\]

### Step 1: Express \(\sec(y)\) in terms of \(\cos(y)\)

We know that \(\sec(y) = \frac{1}{\cos(y)}\), so substitute that into the expression:

\[
\frac{4 + 7 \cos(y)}{7 + 4 \cdot \frac{1}{\cos(y)}}
\]

This simplifies to:

\[
\frac{4 + 7 \cos(y)}{7 + \frac{4}{\cos(y)}}
\]

### Step 2: Eliminate the fraction in the denominator

To simplify the denominator, we can combine it over a common denominator \(\cos(y)\):

\[
\frac{4 + 7 \cos(y)}{\frac{7 \cos(y) + 4}{\cos(y)}}
\]

This is now a complex fraction. To simplify, multiply both the numerator and the denominator by \(\cos(y)\):

\[
\frac{(4 + 7 \cos(y)) \cdot \cos(y)}{7 \cos(y) + 4}
\]

### Step 3: Distribute \(\cos(y)\) in the numerator

Distribute \(\cos(y)\) to both terms in the numerator:

\[
\frac{4 \cos(y) + 7 \cos^2(y)}{7 \cos(y) + 4}
\]

This is the simplified expression.

### Final Answer:
\[
\frac{4 \cos(y) + 7 \cos^2(y)}{7 \cos(y) + 4}
\]

Would you like further details or have any questions?

Here are five related questions for you:
1. What are the values of \(\cos(y)\) for standard angles like 0°, 30°, 45°, and 90°?
2. How do you simplify complex fractions involving trigonometric functions?
3. What are the key relationships between \(\sec(y)\), \(\cos(y)\), and other trigonometric functions?
4. Can this expression be simplified further under certain conditions, like specific values of \(y\)?
5. How do different trigonometric identities assist in simplifying expressions?

**Tip:** When simplifying trigonometric expressions, always check if using fundamental identities can help reduce the complexity.

Simplify the trigonometric expression: (4 + 7cos(y)) / (7 + 4sec(y))

Learn how to simplify the trigonometric expression (4 + 7cos(y)) / (7 + 4sec(y)) using trigonometric identities and algebraic techniques. This problem covers key concepts in trigonometry such as secant, cosine, and simplifying complex fractions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation