Question

If \(\cos\theta = -\frac{3}{5}\) and \(\theta\) does not lie in second quadrant, then \(\tan\frac{\theta}{2} =\)

• A. \(2 \)
• B. \(1 \)
• C. \(-2 \)
• D. \(-1 \)

Hint:

When using half-angle formulas that involve a square root, always determine the correct sign (\(\pm\)) by identifying the quadrant of the half-angle. This requires analyzing the quadrant of the original angle based on the given information.

Answer 1

The Correct Option is C

Step 1: Use the half-angle formula for tangent. We are given \(\cos\theta = -\frac{3}{5}\). We need to find \(\tan\frac{\theta}{2}\). 
The half-angle formula for tangent is given by: \[ \tan\frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \] Substitute the value of \(\cos\theta\): \[ \tan\frac{\theta}{2} = \pm \sqrt{\frac{1 - (-\frac{3}{5})}{1 + (-\frac{3}{5})}} \] \[ = \pm \sqrt{\frac{1 + \frac{3}{5}}{1 - \frac{3}{5}}} \] \[ = \pm \sqrt{\frac{\frac{5+3}{5}}{\frac{5-3}{5}}} \] \[ = \pm \sqrt{\frac{\frac{8}{5}}{\frac{2}{5}}} \] \[ = \pm \sqrt{\frac{8}{2}} \] \[ = \pm \sqrt{4} \] \[ = \pm 2 \] 
Step 2: Determine the sign of \(\tan\frac{\theta}{2}\) based on the quadrant of \(\theta\). We are given that \(\cos\theta = -\frac{3}{5}\). Since cosine is negative, \(\theta\) must lie in either the second quadrant or the third quadrant. The problem states that \(\theta\) does not lie in the second quadrant. 
Therefore, \(\theta\) must lie in the third quadrant. If \(\theta\) is in the third quadrant, its general form can be written as \( (2n\pi + \pi)<\theta<(2n\pi + \frac{3\pi}{2}) \) for any integer \(n\). 
Let's consider the principal range for \(\theta\): \(\pi<\theta<\frac{3\pi}{2}\). Now, divide the inequality by 2 to find the range for \(\frac{\theta}{2}\): \[ \frac{\pi}{2}<\frac{\theta}{2}<\frac{3\pi}{4} \] An angle \(\frac{\theta}{2}\) lying between \(\frac{\pi}{2}\) and \(\frac{3\pi}{4}\) is in the second quadrant.
Step 3: Conclude the value of \(\tan\frac{\theta}{2}\). In the second quadrant, the tangent function is negative. 
Since we found \(\tan\frac{\theta}{2} = \pm 2\), and \(\frac{\theta}{2}\) is in the second quadrant, we must choose the negative sign. 
Therefore, \(\tan\frac{\theta}{2} = -2\). The final answer is -2