If ` 3 sin theta + 4 cos theta =5` , then value of `sin theta ` is

Options

A. `2/3`
B. `4/5`
C. `3/5`
D. `5/3`

Explanation

To solve the equation \( 3 \sin \theta + 4 \cos \theta = 5 \) for \( \sin \theta \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ 3 \sin \theta + 4 \cos \theta = 5 \]
Squaring both sides gives:
\[ (3 \sin \theta + 4 \cos \theta)^2 = 5^2 \]
This simplifies to:
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 \cos^2 \theta = 25 \]

### Step 2: Use the Pythagorean identity
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \). We can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into our equation gives:
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16(1 - \sin^2 \theta) = 25 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 - 16 \sin^2 \theta = 25 \]
Combining like terms results in:
\[ (9 - 16) \sin^2 \theta + 24 \sin \theta \cos \theta + 16 = 25 \]
This simplifies to:
\[ -7 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 - 25 = 0 \]
or
\[ -7 \sin^2 \theta + 24 \sin \theta \cos \theta - 9 = 0 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 7 \sin^2 \theta - 24 \sin \theta \cos \theta + 9 = 0 \]

### Step 5: Solve for \( \sin \theta \)
Now we can use the quadratic formula. Let \( x = \sin \theta \) and \( y = \cos \theta \):
\[ 7x^2 - 24xy + 9 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 7 \), \( b = -24y \), and \( c = 9 \):
\[ x = \frac{24y \pm \sqrt{(-24y)^2 - 4 \cdot 7 \cdot 9}}{2 \cdot 7} \]
Calculating the discriminant:
\[ 576y^2 - 252 = 324y^2 \]
Thus,
\[ x = \frac{24y \pm \sqrt{324y^2}}{14} \]
This simplifies to:
\[ x = \frac{24y \pm 18y}{14} \]
This gives two cases:
1. \( x = \frac{42y}{14} = 3y \)
2. \( x = \frac{6y}{14} = \frac{3y}{7} \)

### Step 6: Substitute back to find \( \sin \theta \)
Using the original equation \( 3 \sin \theta + 4 \cos \theta = 5 \):
1. For \( \sin \theta = 3 \cos \theta \):
\[ 3(3 \cos \theta) + 4 \cos \theta = 5 \implies 9 \cos \theta + 4 \cos \theta = 5 \implies 13 \cos \theta = 5 \implies \cos \theta = \frac{5}{13} \]
Then,
\[ \sin \theta = 3 \cdot \frac{5}{13} = \frac{15}{13} \text{ (not possible since } \sin \theta \leq 1\text{)} \]

2. For \( \sin \theta = \frac{3}{7} \cos \theta \):
Substitute into the original equation:
\[ 3 \left(\frac{3}{7} \cos \theta\right) + 4 \cos \theta = 5 \implies \frac{9}{7} \cos \theta + 4 \cos \theta = 5 \]
Combine:
\[ \left(\frac{9}{7} + \frac{28}{7}\right) \cos \theta = 5 \implies \frac{37}{7} \cos \theta = 5 \implies \cos \theta = \frac{35}{37} \]
Then,
\[ \sin \theta = \frac{3}{7} \cdot \frac{35}{37} = \frac{105}{259} \]

### Final Answer
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{3}{5} \]