Solving Trigonometric Expressions with Given Tangent

Introduction

In trigonometry, we often need to solve complex expressions involving trigonometric functions, especially when given the value of a tangent. This article demonstrates how to solve the expression frac{4 sin A - cos A}{3 cos A sin A} given that tan A frac{4}{5}. We'll explore multiple methods and consider different quadrants to ensure a comprehensive solution.

Solution Method 1: Using Trigonometric Identities

Given tan A frac{4}{5}, we can represent sin A and cos A in terms of a right triangle. Let the opposite side be 4 and the adjacent side be 5. The hypotenuse h can be calculated using the Pythagorean theorem:

h sqrt{4^2 5^2} sqrt{16 25} sqrt{41}

So, we can find:

sin A frac{4}{sqrt{41}} cos A frac{5}{sqrt{41}}

Next, let's substitute these into the expression:

4 sin A 4 cdot frac{4}{sqrt{41}} frac{16}{sqrt{41}} 3 cos A 3 cdot frac{5}{sqrt{41}} frac{15}{sqrt{41}}

Hence:

4 sin A - cos A frac{16}{sqrt{41}} - frac{5}{sqrt{41}} frac{16 - 5}{sqrt{41}} frac{11}{sqrt{41}}

3 cos A sin A frac{15}{sqrt{41}} cdot frac{4}{sqrt{41}} frac{15 cdot 4}{41} frac{19}{sqrt{41}}

Thus, the expression simplifies to:

frac{4 sin A - cos A}{3 cos A sin A} frac{frac{11}{sqrt{41}}}{frac{19}{sqrt{41}}} frac{11}{19}

Solution Method 2: Algebraic Manipulation

Another approach is to manipulate the expression directly:

Given: tan A frac{4}{5}, rewrite the expression as:

frac{4 sin A - cos A}{3 cos A sin A} [4 sin A / cos A - cos A / cos A] / [3 cos A / cos A * sin A / cos A] 4 tan A - 1 / [3 * tan A]

Evaluate:

[4 * frac{4}{5} - 1] / [3 * frac{4}{5}] [frac{16}{5} - 1] / 3 * frac{4}{5} [frac{11}{5}] / [frac{12}{5}] frac{11}{19}

Considering Quadrant Factors

In trigonometry, it's important to consider the quadrant where angle A lies, as sine and cosine values can be positive or negative in different quadrants. From the given tangent, we know:

A could be in the 1st or 3rd quadrant.

1st Quadrant (0 to 90 degrees)

In the 1st quadrant, both sine and cosine are positive. Radius: sqrt{41} gives us:

sin A frac{4}{sqrt{41}} cos A frac{5}{sqrt{41}}

The steps followed are the same as Method 1, leading to:

frac{4 sin A - cos A}{3 cos A sin A} frac{frac{11}{sqrt{41}}}{frac{19}{sqrt{41}}} frac{11}{19}

3rd Quadrant (180 to 270 degrees)

In the 3rd quadrant, both sine and cosine are negative. Radius: sqrt{41} gives us:

sin A -frac{4}{sqrt{41}} cos A -frac{5}{sqrt{41}}

The steps are similar but with negative values:

1. 4 sin A 4 cdot -frac{4}{sqrt{41}} -frac{16}{sqrt{41}}

2. 3 cos A 3 cdot -frac{5}{sqrt{41}} -frac{15}{sqrt{41}}

3. 4 sin A - cos A -frac{16}{sqrt{41}} - (-frac{5}{sqrt{41}}) -frac{11}{sqrt{41}}

4. 3 cos A sin A -frac{15}{sqrt{41}} cdot -frac{4}{sqrt{41}} frac{19}{sqrt{41}}

5. frac{4 sin A - cos A}{3 cos A sin A} frac{frac{-11}{sqrt{41}}}{frac{19}{sqrt{41}}} -frac{11}{19}

The final result for the 3rd quadrant is -frac{11}{19}.

Conclusion

We have shown that the expression frac{4 sin A - cos A}{3 cos A sin A}, given tan A frac{4}{5}, simplifies to frac{11}{19} in the 1st quadrant. For the 3rd quadrant, the result is -frac{11}{19}. This demonstrates the importance of considering the quadrant when solving trigonometric expressions.