If 8sinx- 4 = cos.x, the values of sinx 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 \( 8 \sin x - 4 = \cos x \) for the values of \( \sin x \), follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ 8 \sin x - 4 = \cos x \]
We can rearrange this to express everything in terms of \( \sin x \):
\[ 8 \sin x - \cos x - 4 = 0 \]

### Step 2: Square both sides
To eliminate the cosine term, we square both sides:
\[ (8 \sin x - 4)^2 = \cos^2 x \]
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we rewrite the equation:
\[ (8 \sin x - 4)^2 = 1 - \sin^2 x \]

### Step 3: Expand the left side
Now, we expand the left side:
\[ (8 \sin x - 4)(8 \sin x - 4) = 64 \sin^2 x - 64 \sin x + 16 \]
So, we have:
\[ 64 \sin^2 x - 64 \sin x + 16 = 1 - \sin^2 x \]

### Step 4: Rearrange the equation
Now, we bring all terms 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: Solve the quadratic equation
Let \( t = \sin x \). The equation becomes:
\[ 65t^2 - 64t + 15 = 0 \]
We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 65, b = -64, c = 15 \).

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

### Step 7: Apply the quadratic formula
Now, substitute back into the quadratic formula:
\[ t = \frac{64 \pm \sqrt{196}}{2 \cdot 65} \]
Since \( \sqrt{196} = 14 \), we have:
\[ t = \frac{64 \pm 14}{130} \]

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

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