If `tan theta+sec theta=1.5`, then value of `sin theta` is :

Options

A. `(5)/(13)`
B. `(15)/(13)`
C. `(3)/(5)`
D. `(2)/(3)`

Answer

To solve the equation \( \tan \theta + \sec \theta = 1.5 \) and find the value of \( \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} = 1.5 \]
Combining the terms gives:
\[ \frac{\sin \theta + 1}{\cos \theta} = 1.5 \]

### Step 2: Clear the fraction
To eliminate the fraction, multiply both sides by \( \cos \theta \):
\[ \sin \theta + 1 = 1.5 \cos \theta \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ \sin \theta = 1.5 \cos \theta - 1 \]

### Step 4: Square both sides
Now we square both sides to eliminate the sine:
\[ \sin^2 \theta = (1.5 \cos \theta - 1)^2 \]
Expanding the right side:
\[ \sin^2 \theta = (1.5^2 \cos^2 \theta - 2 \cdot 1.5 \cdot 1 \cdot \cos \theta + 1^2) \]
\[ \sin^2 \theta = 2.25 \cos^2 \theta - 3 \cos \theta + 1 \]

### Step 5: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can replace \( \sin^2 \theta \):
\[ 1 - \cos^2 \theta = 2.25 \cos^2 \theta - 3 \cos \theta + 1 \]
This simplifies to:
\[ 0 = 3.25 \cos^2 \theta - 3 \cos \theta \]

### Step 6: Factor the equation
Factoring gives:
\[ \cos \theta (3.25 \cos \theta - 3) = 0 \]
This results in two cases:
1. \( \cos \theta = 0 \)
2. \( 3.25 \cos \theta - 3 = 0 \)

### Step 7: Solve for \( \cos \theta \)
From the second case:
\[ 3.25 \cos \theta = 3 \implies \cos \theta = \frac{3}{3.25} = \frac{12}{13} \]

### Step 8: Find \( \sin \theta \)
Using the Pythagorean identity:
\[ \sin^2 \theta + \left(\frac{12}{13}\right)^2 = 1 \]
Calculating \( \left(\frac{12}{13}\right)^2 \):
\[ \sin^2 \theta + \frac{144}{169} = 1 \]
\[ \sin^2 \theta = 1 - \frac{144}{169} = \frac{25}{169} \]
Taking the square root:
\[ \sin \theta = \frac{5}{13} \quad \text{(since sine is positive in the first quadrant)} \]

### Final Answer
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{5}{13} \]