If `tan theta = 4/3`, what is the value of `sqrt((1-sintheta)/(1+sintheta))`

Answer

To solve the problem where \( \tan \theta = \frac{4}{3} \) and we need to find the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \), we can follow these steps:

### Step 1: Understand the relationship of tangent
Given \( \tan \theta = \frac{4}{3} \), we can interpret this as:
- Opposite side (perpendicular) = 4
- Adjacent side (base) = 3

### Step 2: Find the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(\text{Opposite})^2 + (\text{Adjacent})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Calculate \( \sin \theta \)
Now we can find \( \sin \theta \):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5} \]

### Step 4: Substitute \( \sin \theta \) into the expression
Now we substitute \( \sin \theta \) into the expression \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \):
\[ \sqrt{\frac{1 - \frac{4}{5}}{1 + \frac{4}{5}}} \]

### Step 5: Simplify the expression
Calculating the numerator and denominator:
- Numerator:
\[ 1 - \frac{4}{5} = \frac{5}{5} - \frac{4}{5} = \frac{1}{5} \]
- Denominator:
\[ 1 + \frac{4}{5} = \frac{5}{5} + \frac{4}{5} = \frac{9}{5} \]

Now substituting these values back:
\[ \sqrt{\frac{\frac{1}{5}}{\frac{9}{5}}} = \sqrt{\frac{1}{5} \cdot \frac{5}{9}} = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \) is \( \frac{1}{3} \).

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