If A lies in the second quadrant and `3tanA+4=0`, then the value of `2cotA-5cosA+sinA` is

Options

A. (a)`-53/(10)`
B. (b)`23/(10)`
C. (c)`37/(10)`
D. (d)`7/(10)`

Answer

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

### Step 1: Given Information
We are given that \( A \) lies in the second quadrant and the equation \( 3 \tan A + 4 = 0 \).

### Step 2: Solve for \( \tan A \)
From the equation \( 3 \tan A + 4 = 0 \), we can isolate \( \tan A \):
\[ 3 \tan A = -4 \implies \tan A = -\frac{4}{3} \]

### Step 3: Identify the Sides of the Triangle
Since \( \tan A = \frac{\text{perpendicular}}{\text{base}} \), we can assign:
- Perpendicular = 4
- Base = 3

### Step 4: Calculate the Hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(\text{perpendicular})^2 + (\text{base})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 5: Find \( \cot A \)
Since \( \cot A = \frac{\text{base}}{\text{perpendicular}} \):
\[ \cot A = \frac{3}{4} \]
However, in the second quadrant, \( \cot A \) is negative:
\[ \cot A = -\frac{3}{4} \]

### Step 6: Find \( \sin A \) and \( \cos A \)
Using the definitions:
- \( \sin A = \frac{\text{perpendicular}}{\text{hypotenuse}} = \frac{4}{5} \) (positive in the second quadrant)
- \( \cos A = \frac{\text{base}}{\text{hypotenuse}} = \frac{3}{5} \) (negative in the second quadrant):
\[ \cos A = -\frac{3}{5} \]

### Step 7: Substitute Values into the Expression
We need to find the value of \( 2 \cot A - 5 \cos A + \sin A \):
\[ 2 \cot A - 5 \cos A + \sin A = 2 \left(-\frac{3}{4}\right) - 5 \left(-\frac{3}{5}\right) + \frac{4}{5} \]

### Step 8: Simplify the Expression
Calculating each term:
\[ 2 \cot A = -\frac{3}{2} \]
\[ -5 \cos A = 3 \]
Now substituting these values:
\[ -\frac{3}{2} + 3 + \frac{4}{5} \]

### Step 9: Find a Common Denominator
The common denominator for \( 2 \) and \( 5 \) is \( 10 \):
\[ -\frac{3}{2} = -\frac{15}{10}, \quad 3 = \frac{30}{10}, \quad \frac{4}{5} = \frac{8}{10} \]
Now substituting these values:
\[ -\frac{15}{10} + \frac{30}{10} + \frac{8}{10} = \frac{-15 + 30 + 8}{10} = \frac{23}{10} \]

### Final Answer
The value of \( 2 \cot A - 5 \cos A + \sin A \) is:
\[ \frac{23}{10} \]