If `tantheta-cottheta=(119)/(60)" for "0^(@)ltthetaltpi//2`, then the value of `sintheta+costheta`.

Options

A. `(17)/(13)`
B. `(21)/(13)`
C. `(19)/(13)`
D. `(23)/(13)`

Answer

To solve the equation \( \tan \theta - \cot \theta = \frac{119}{60} \) for \( 0 < \theta < \frac{\pi}{2} \) and find the value of \( \sin \theta + \cos \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \cot \theta = \frac{1}{\tan \theta} \). So we can rewrite the equation as:
\[ \tan \theta - \frac{1}{\tan \theta} = \frac{119}{60} \]

### Step 2: Multiply through by \( \tan \theta \)
To eliminate the fraction, we multiply both sides by \( \tan \theta \):
\[ \tan^2 \theta - 1 = \frac{119}{60} \tan \theta \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \tan^2 \theta - \frac{119}{60} \tan \theta - 1 = 0 \]

### Step 4: Let \( x = \tan \theta \)
Let \( x = \tan \theta \). The equation becomes:
\[ x^2 - \frac{119}{60} x - 1 = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -\frac{119}{60}, c = -1 \):
\[ x = \frac{\frac{119}{60} \pm \sqrt{\left(\frac{119}{60}\right)^2 + 4}}{2} \]

### Step 6: Simplify the discriminant
Calculating the discriminant:
\[ \left(\frac{119}{60}\right)^2 + 4 = \frac{14161}{3600} + \frac{14400}{3600} = \frac{28561}{3600} \]
Taking the square root:
\[ \sqrt{\frac{28561}{3600}} = \frac{169}{60} \]

### Step 7: Substitute back into the formula
Now substituting back into the quadratic formula:
\[ x = \frac{\frac{119}{60} \pm \frac{169}{60}}{2} \]
This gives us two possible values:
\[ x = \frac{288}{120} \quad \text{or} \quad x = \frac{-50}{120} \]
Since \( x = \tan \theta \) must be positive in the interval \( 0 < \theta < \frac{\pi}{2} \), we take:
\[ x = \frac{288}{120} = \frac{12}{5} \]

### Step 8: Find \( \sin \theta \) and \( \cos \theta \)
Using the triangle ratios, we have:
- Opposite side = 12
- Adjacent side = 5
- Hypotenuse = \( \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = 13 \)

Thus:
\[ \sin \theta = \frac{12}{13}, \quad \cos \theta = \frac{5}{13} \]

### Step 9: Calculate \( \sin \theta + \cos \theta \)
Now we can find:
\[ \sin \theta + \cos \theta = \frac{12}{13} + \frac{5}{13} = \frac{12 + 5}{13} = \frac{17}{13} \]

### Final Answer
The value of \( \sin \theta + \cos \theta \) is:
\[ \frac{17}{13} \]