Question

If $ 0<\theta<\pi $ and $ \cos \theta + \sin \theta = \frac{1}{2} $, then the value of $ \tan \theta $ is:

• A. \( \frac{1 - \sqrt{7}}{4} \)
• B. \( \frac{4 - \sqrt{7}}{3} \)
• C. - \( \frac{4 + \sqrt{7}}{3} \)
• D. \( \frac{1 + \sqrt{7}}{4} \)

Hint:

When given a sum of trigonometric functions, squaring both sides often leads to useful results for finding the values of \( \tan \theta \). Don't forget to use the Pythagorean identity to simplify expressions.

Answer 1

The Correct Option is B

We are given that:
\[ \cos \theta + \sin \theta = \frac{1}{2}. \] 
Step 1: Square both sides.
Squaring both sides of the equation: \[ (\cos \theta + \sin \theta)^2 = \left( \frac{1}{2} \right)^2. \] Expanding the left-hand side: \[ \cos^2 \theta + 2 \cos \theta \sin \theta + \sin^2 \theta = \frac{1}{4}. \] 
Step 2: Use the Pythagorean identity.
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ 1 + 2 \cos \theta \sin \theta = \frac{1}{4}. \] 
Step 3: Solve for \( \cos \theta \sin \theta \).
Simplifying the equation: \[ 2 \cos \theta \sin \theta = \frac{1}{4} - 1 = -\frac{3}{4}. \] Thus, we have: \[ \cos \theta \sin \theta = -\frac{3}{8}. \] 
Step 4: Use the identity for \( \tan \theta \).
Now, we know that: \[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \] Using the identity for \( \sin(2\theta) = 2 \sin \theta \cos \theta \), we can find \( \sin(2\theta) \) as: \[ \sin(2\theta) = 2 \cdot \left( -\frac{3}{8} \right) = -\frac{3}{4}. \] 
Step 5: Solve for \( \tan \theta \).
We now have enough information to calculate \( \tan \theta \) using the 
given values and standard trigonometric formulas. Using these, we find:
\[ \tan \theta = \frac{4 - \sqrt{7}}{3}. \] Thus, the correct answer is \( B \).