If `8 sin x = 4 + cos x`, then the values of sin x are

Options

A. `(3)/(5), (-5)/(13)`
B. `(-3)/(5), (-5)/(13)`
C. `(3)/(5), (5)/(13)`
D. `(5)/(3), (5)/(13)`

Answer

To solve the equation \( 8 \sin x = 4 + \cos x \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \(\cos x\):
\[ 8 \sin x - 4 = \cos x \]

### Step 2: Using the Pythagorean Identity
We know that \(\cos^2 x + \sin^2 x = 1\). Therefore, we can express \(\cos x\) in terms of \(\sin x\):
\[ \cos x = \sqrt{1 - \sin^2 x} \]
Substituting this into our rearranged equation gives:
\[ 8 \sin x - 4 = \sqrt{1 - \sin^2 x} \]

### Step 3: Squaring Both Sides
To eliminate the square root, we square both sides:
\[ (8 \sin x - 4)^2 = 1 - \sin^2 x \]
Expanding the left side:
\[ 64 \sin^2 x - 64 \sin x + 16 = 1 - \sin^2 x \]

### Step 4: Rearranging to Form a Quadratic Equation
Now, we rearrange the equation to one side:
\[ 64 \sin^2 x + \sin^2 x - 64 \sin x + 16 - 1 = 0 \]
This simplifies to:
\[ 65 \sin^2 x - 64 \sin x + 15 = 0 \]

### Step 5: Applying the Quadratic Formula
Now we can apply the quadratic formula \( \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 65 \), \( b = -64 \), and \( c = 15 \):
\[ \sin x = \frac{64 \pm \sqrt{(-64)^2 - 4 \cdot 65 \cdot 15}}{2 \cdot 65} \]

### Step 6: Calculating the Discriminant
Calculating the discriminant:
\[ (-64)^2 = 4096 \]
\[ 4 \cdot 65 \cdot 15 = 3900 \]
Thus,
\[ b^2 - 4ac = 4096 - 3900 = 196 \]

### Step 7: Continuing with the Quadratic Formula
Now substituting back into the formula:
\[ \sin x = \frac{64 \pm \sqrt{196}}{130} \]
Since \(\sqrt{196} = 14\), we have:
\[ \sin x = \frac{64 \pm 14}{130} \]

### Step 8: Finding the Two Values
Calculating the two possible values:
1. \( \sin x = \frac{64 + 14}{130} = \frac{78}{130} = \frac{39}{65} \)
2. \( \sin x = \frac{64 - 14}{130} = \frac{50}{130} = \frac{5}{13} \)

### Final Answer
Thus, the values of \(\sin x\) are:
\[ \sin x = \frac{39}{65} \quad \text{and} \quad \sin x = \frac{5}{13} \]