Take $ as tita,

a sin $ − b cos $ = c ,then a + b tan $ ÷ a tan − b = ?

To solve the equation \( a \sin \theta - b \cos \theta = c \) and find the expression \( a + \frac{b \tan \theta}{a \tan \theta - b} \), we can break it down step by step. Let's start by rewriting the equation in a more manageable form.

Rearranging the Equation

We have the equation:

a sin θ - b cos θ = c

From this, we can express \( \sin \theta \) and \( \cos \theta \) in terms of \( c \), \( a \), and \( b \). However, for our purpose, we will focus on manipulating the expression we need to evaluate.

Understanding the Expression

The expression we want to evaluate is:

a + \frac{b \tan \theta}{a \tan \theta - b}

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means we can rewrite \( b \tan \theta \) as:

b \tan \theta = b \frac{\sin \theta}{\cos \theta}

Substituting for Tangent

Now, substituting \( \tan \theta \) into our expression gives:

a + \frac{b \frac{\sin \theta}{\cos \theta}}{a \frac{\sin \theta}{\cos \theta} - b}

This simplifies to:

a + \frac{b \sin \theta}{a \sin \theta - b \cos \theta}

Using the Original Equation

From our original equation \( a \sin \theta - b \cos \theta = c \), we can substitute \( c \) into our expression:

a + \frac{b \sin \theta}{c}

Final Simplification

Now, we can combine the terms. The expression becomes:

a + \frac{b \sin \theta}{c}

To find a common denominator, we can express \( a \) as \( \frac{ac}{c} \):

\frac{ac + b \sin \theta}{c}

Conclusion

Thus, the final result for the expression \( a + \frac{b \tan \theta}{a \tan \theta - b} \) simplifies to:

\frac{ac + b \sin \theta}{c}

This shows how the original equation relates to the expression we were trying to evaluate. By substituting and simplifying, we can see the connection clearly.