If `3 cos^2 A+7 sin^2 A=4` then what is the value of cot A. given that A is an acute angle

Options

A. 1
B. `sqrt3`
C. `sqrt3/2`
D. `1/sqrt3`

Answer

To solve the equation \(3 \cos^2 A + 7 \sin^2 A = 4\) and find the value of \(\cot A\) given that \(A\) is an acute angle, we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3 \cos^2 A + 7 \sin^2 A = 4 \]
We know that \(\cos^2 A + \sin^2 A = 1\). We can express \(\cos^2 A\) in terms of \(\sin^2 A\):
\[ \cos^2 A = 1 - \sin^2 A \]
Substituting this into the equation gives:
\[ 3(1 - \sin^2 A) + 7 \sin^2 A = 4 \]

### Step 2: Simplify the equation
Expanding this, we have:
\[ 3 - 3 \sin^2 A + 7 \sin^2 A = 4 \]
Combining like terms:
\[ 3 + 4 \sin^2 A = 4 \]

### Step 3: Isolate \(\sin^2 A\)
Now, we isolate \(\sin^2 A\):
\[ 4 \sin^2 A = 4 - 3 \]
\[ 4 \sin^2 A = 1 \]
Dividing both sides by 4:
\[ \sin^2 A = \frac{1}{4} \]

### Step 4: Find \(\sin A\)
Taking the square root of both sides, we find:
\[ \sin A = \frac{1}{2} \]
Since \(A\) is an acute angle, we take the positive root.

### Step 5: Find \(\cos A\)
Using the identity \(\cos^2 A + \sin^2 A = 1\):
\[ \cos^2 A = 1 - \sin^2 A = 1 - \frac{1}{4} = \frac{3}{4} \]
Taking the square root:
\[ \cos A = \frac{\sqrt{3}}{2} \]

### Step 6: Calculate \(\cot A\)
Now, we can find \(\cot A\):
\[ \cot A = \frac{\cos A}{\sin A} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

Thus, the value of \(\cot A\) is:
\[ \cot A = \sqrt{3} \]

### Final Answer
\(\cot A = \sqrt{3}\)

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