If A lies between `180^@` and `270^@` and 3 tan A=4 , find the value of 2 cot A - cos A + sin A

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Identify the given information
We know that \( A \) lies between \( 180^\circ \) and \( 270^\circ \) and that \( 3 \tan A = 4 \).

### Step 2: Find the value of \( \tan A \)
From the equation \( 3 \tan A = 4 \), we can find \( \tan A \):
\[ \tan A = \frac{4}{3} \]

### Step 3: Determine the values of sine and cosine
Since \( A \) is in the third quadrant, both sine and cosine will be negative, while tangent will be positive. We can represent \( \tan A \) as:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \]
Let the opposite side (perpendicular) be 4 and the adjacent side (base) be 3.

### Step 4: Calculate the hypotenuse using Pythagoras theorem
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
\[ \text{hypotenuse}^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
\[ \text{hypotenuse} = \sqrt{25} = 5 \]

### Step 5: Find \( \sin A \) and \( \cos A \)
Now we can find \( \sin A \) and \( \cos A \):
\[ \sin A = -\frac{\text{opposite}}{\text{hypotenuse}} = -\frac{4}{5} \]
\[ \cos A = -\frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{3}{5} \]

### Step 6: Find \( \cot A \)
We can find \( \cot A \) as follows:
\[ \cot A = \frac{1}{\tan A} = \frac{3}{4} \]

### Step 7: Substitute values into the expression \( 2 \cot A - \cos A + \sin A \)
Now we substitute the values into the expression:
\[ 2 \cot A - \cos A + \sin A = 2 \left(\frac{3}{4}\right) - \left(-\frac{3}{5}\right) + \left(-\frac{4}{5}\right) \]
\[ = \frac{6}{4} + \frac{3}{5} - \frac{4}{5} \]
\[ = \frac{3}{2} + \frac{3}{5} - \frac{4}{5} \]
\[ = \frac{3}{2} + \frac{3 - 4}{5} \]
\[ = \frac{3}{2} - \frac{1}{5} \]

### Step 8: Find a common denominator and simplify
The common denominator between 2 and 5 is 10:
\[ = \frac{3 \times 5}{10} - \frac{1 \times 2}{10} = \frac{15}{10} - \frac{2}{10} = \frac{13}{10} \]

### Final Answer
Thus, the value of \( 2 \cot A - \cos A + \sin A \) is:
\[ \frac{13}{10} \]

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