Solving Trigonometric Expressions Using Tangent and Pythagorean Theorem

In this article, we will solve the trigonometric expression frac{2sin A - 3cos A}{2sin A 3cos A} given that tan A frac{4}{3}. We'll explore the use of the tangent value, the Pythagorean theorem, and the trigonometric identities to find the solution step-by-step.

Introduction to Tangent, Sine, and Cosine

Trigonometry is a branch of mathematics concerned with the relationships between the angles and sides of triangles. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The sine and cosine functions are also useful in determining the ratios of the sides of a right triangle. Given tan A frac{4}{3}, we can represent this as the opposite side being 4 and the adjacent side being 3.

Applying the Pythagorean Theorem

The Pythagorean theorem states that the square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides. Using this theorem, we can find the hypotenuse H of the triangle:

[H sqrt{4^2 3^2} sqrt{16 9} sqrt{25} 5]

Calculating Sine and Cosine

Now that we have the hypotenuse, we can find the sine and cosine of A using their definitions:

[sin A frac{text{opposite}}{text{hypotenuse}} frac{4}{5}] [cos A frac{text{adjacent}}{text{hypotenuse}} frac{3}{5}]

Evaluating the Expression

We need to solve the expression frac{2sin A - 3cos A}{2sin A 3cos A}. To do this, we can substitute the values of sine and cosine that we calculated:

[2sin A 2 cdot frac{4}{5} frac{8}{5}] [3cos A 3 cdot frac{3}{5} frac{9}{5}]

Substitute these into the expression:

[text{{Numerator}}: 2sin A - 3cos A frac{8}{5} - frac{9}{5} frac{8 - 9}{5} frac{-1}{5}] [text{{Denominator}}: 2sin A 3cos A frac{8}{5} frac{9}{5} frac{8 9}{5} frac{17}{5}]

Therefore, the expression simplifies to:

[frac{2sin A - 3cos A}{2sin A 3cos A} frac{frac{-1}{5}}{frac{17}{5}} frac{-1}{5} cdot frac{5}{17} frac{-1}{17}]

Thus, the value of the expression is -frac{1}{17}.

Alternative Method Using Tangent and Reciprocal Identities

Here is an alternative method to solve the expression using the tangent, secant, and identity properties:

[1 tan^2 A sec^2 A] [1 left(frac{4}{3}right)^2 sec^2 A] [1 frac{16}{9} sec^2 A] [frac{25}{9} sec^2 A] [sec A frac{5}{3}] [cos A frac{1}{sec A} frac{3}{5}] [sin A frac{4}{5}] [2sin A - 3cos A 2 cdot frac{4}{5} - 3 cdot frac{3}{5} frac{8}{5} - frac{9}{5} frac{-1}{5}] [2sin A 3cos A 2 cdot frac{4}{5} 3 cdot frac{3}{5} frac{8}{5} frac{9}{5} frac{17}{5}] [frac{2sin A - 3cos A}{2sin A 3cos A} frac{frac{-1}{5}}{frac{17}{5}} frac{-1}{5} cdot frac{5}{17} -frac{1}{17}]

In both methods, we arrive at the same result, confirming our solution.

Understanding and applying the Tangent and Pythagorean Theorem is essential in solving complex trigonometric expressions. This article provides a detailed breakdown of the process, offering a step-by-step guide to solving similar problems.

Conclusion

This article demonstrated the solution to the trigonometric expression using both direct substitution and alternative methods involving identities and reciprocal functions. Mastery of these techniques is critical for students and professionals in mathematics, engineering, and other fields requiring advanced trigonometric knowledge.