Understanding the Trigonometry Problem

The question asks us to find the value of a specific trigonometric expression involving \(\sin\theta\) and \(\cos\theta\), given a condition involving \(\tan\theta\). The given condition is \(3 \tan \theta = 2\), and the expression to evaluate is \(\frac{2 \sin\theta-\cos\theta}{2 \cos\theta-\sin\theta}\).

Analyzing the Given Condition

The condition \(3 \tan \theta = 2\) can be easily rewritten to find the value of \(\tan\theta\).

From \(3 \tan \theta = 2\), we get:

\[ \tan \theta = \frac{2}{3} \]

This value of \(\tan\theta\) is key to solving the problem.

Evaluating the Trigonometric Expression

We need to find the value of the expression \(\frac{2 \sin\theta-\cos\theta}{2 \cos\theta-\sin\theta}\). Notice that this expression contains both \(\sin\theta\) and \(\cos\theta\). Since we know the value of \(\tan\theta\) and we know that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), we can transform the given expression into terms of \(\tan\theta\). This is a common technique when dealing with expressions involving a mix of \(\sin\theta\) and \(\cos\theta\) and a known value for \(\tan\theta\).

To do this, we can divide both the numerator and the denominator of the expression by \(\cos\theta\). We must be careful here, assuming \(\cos\theta \ne 0\). If \(\cos\theta = 0\), then \(\theta\) would be \(90^\circ\) or \(270^\circ\) (or \(90^\circ + n \cdot 180^\circ\)), making \(\tan\theta\) undefined. Since \(\tan\theta\) is given as \(\frac{2}{3}\), \(\cos\theta\) cannot be zero.

Step-by-Step Transformation:

The expression is: \(\frac{2 \sin\theta-\cos\theta}{2 \cos\theta-\sin\theta}\)

Divide the numerator by \(\cos\theta\):

\[ \frac{2 \sin\theta-\cos\theta}{\cos\theta} = \frac{2 \sin\theta}{\cos\theta} - \frac{\cos\theta}{\cos\theta} = 2 \tan\theta - 1 \]

Divide the denominator by \(\cos\theta\):

\[ \frac{2 \cos\theta-\sin\theta}{\cos\theta} = \frac{2 \cos\theta}{\cos\theta} - \frac{\sin\theta}{\cos\theta} = 2 - \tan\theta \]

So, the original expression can be rewritten as:

\[ \frac{2 \tan\theta - 1}{2 - \tan\theta} \]

Substituting the Value of tanθ:

We found earlier that \(\tan \theta = \frac{2}{3}\). Now, substitute this value into the transformed expression:

\[ \text{Value} = \frac{2 \left(\frac{2}{3}\right) - 1}{2 - \left(\frac{2}{3}\right)} \]

Perform the calculations:

\[ \text{Value} = \frac{\frac{4}{3} - 1}{2 - \frac{2}{3}} \]

To simplify the numerator and denominator, find a common denominator (which is 3):

\[ \text{Value} = \frac{\frac{4}{3} - \frac{3}{3}}{\frac{6}{3} - \frac{2}{3}} \]

\[ \text{Value} = \frac{\frac{4-3}{3}}{\frac{6-2}{3}} \]

\[ \text{Value} = \frac{\frac{1}{3}}{\frac{4}{3}} \]

To divide the fractions, multiply the numerator fraction by the reciprocal of the denominator fraction:

\[ \text{Value} = \frac{1}{3} \times \frac{3}{4} \]

\[ \text{Value} = \frac{1 \times 3}{3 \times 4} = \frac{3}{12} \]

Simplify the fraction:

\[ \text{Value} = \frac{1}{4} \]

Therefore, the value of the expression \(\frac{2 \sin\theta-\cos\theta}{2 \cos\theta-\sin\theta}\) is \(\frac{1}{4}\).

Summary of Steps to Evaluate Trigonometric Expression

1. Use the given condition \(3 \tan \theta = 2\) to find the value of \(\tan\theta\).
2. Recognize that the expression involves \(\sin\theta\) and \(\cos\theta\) and can be simplified using the value of \(\tan\theta\).
3. Divide the numerator and denominator of the expression by \(\cos\theta\) to rewrite it in terms of \(\tan\theta\).
4. Substitute the known value of \(\tan\theta\) into the new expression.
5. Perform the arithmetic calculations to find the final value.

Revision Table: Key Trigonometric Concepts

Additional Information on Evaluating Trig Expressions

When you are given a relationship like \(a \tan \theta = b\) or \(a \cot \theta = b\) and asked to evaluate an expression that is a ratio of linear combinations of \(\sin\theta\) and \(\cos\theta\), dividing the numerator and denominator by \(\cos\theta\) (for \(\tan\theta\)) or \(\sin\theta\) (for \(\cot\theta\)) is a very effective technique. This method works because it transforms the expression into a form that depends only on \(\tan\theta\) or \(\cot\theta\), whose value you can find from the given condition.

For example, an expression like \(\frac{a \sin\theta + b \cos\theta}{c \sin\theta + d \cos\theta}\) can be rewritten by dividing numerator and denominator by \(\cos\theta\) as \(\frac{a \tan\theta + b}{c \tan\theta + d}\), provided \(\cos\theta \ne 0\). Similarly, dividing by \(\sin\theta\) gives \(\frac{a + b \cot\theta}{c + d \cot\theta}\), provided \(\sin\theta \ne 0\).

This technique is useful in many trigonometry problems and helps simplify complex expressions quickly.