Trigonometric Value Calculation

We are given the equation $3 \cot A = 4$. We need to find the value of the expression $(3 \sin A - 4 \cos A)$.

Deriving Trigonometric Ratios

From the given equation, we can find the value of $\cot A$:

$ \cot A = \frac{4}{3} $

Recall that $\cot A = \frac{\text{Adjacent}}{\text{Opposite}}$. We can model this using a right-angled triangle where the adjacent side is $4k$ and the opposite side is $3k$, for some positive constant $k$.

Using the Pythagorean theorem, we find the hypotenuse ($h$):

$ h^2 = (\text{Adjacent})^2 + (\text{Opposite})^2 $

$ h^2 = (4k)^2 + (3k)^2 $

$ h^2 = 16k^2 + 9k^2 $

$ h^2 = 25k^2 $

$ h = \sqrt{25k^2} = 5k $

Now, we can find $\sin A$ and $\cos A$:

• $ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3k}{5k} = \frac{3}{5} $
• $ \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4k}{5k} = \frac{4}{5} $

Evaluating the Expression

Substitute the values of $\sin A$ and $\cos A$ into the expression $(3 \sin A - 4 \cos A)$:

$ (3 \sin A - 4 \cos A) = 3 \left( \frac{3}{5} \right) - 4 \left( \frac{4}{5} \right) $

$ = \frac{9}{5} - \frac{16}{5} $

$ = \frac{9 - 16}{5} $

$ = \frac{-7}{5} $

Thus, the value of the expression is $\frac{-7}{5}$.