Question

If \(\theta \isin(-\pi,0)\ and\ cos\theta=\frac{-12}{13}, then\ \sin(\frac{\theta}{2})=\)

• A. \(\frac{-5\sqrt{26}}{26}\)
• B. \(\frac{5\sqrt{26}}{26}\)
• C. \(\frac{-5\sqrt{13}}{13}\)
• D. \(\frac{5\sqrt{13}}{13}\)
• E. \(\frac{-5\sqrt{13}}{26}\)

Answer 1

The Correct Option is A

We are given: \[ \cos \theta = \frac{-12}{13}, \quad \theta \in (-\pi, 0). \] We need to find \( \sin \left( \frac{\theta}{2} \right) \). We will use the half-angle identity for sine: \[ \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}. \] Since \( \theta \in (-\pi, 0) \), we are in the third quadrant, where sine is negative. Thus, we will take the negative sign in the half-angle formula: \[ \sin \left( \frac{\theta}{2} \right) = -\sqrt{\frac{1 - \cos \theta}{2}}. \] Step 1: Substitute \( \cos \theta = \frac{-12}{13} \) \[ \sin \left( \frac{\theta}{2} \right) = -\sqrt{\frac{1 - \left( \frac{-12}{13} \right)}{2}}. \] Simplifying the expression inside the square root: \[ \sin \left( \frac{\theta}{2} \right) = -\sqrt{\frac{1 + \frac{12}{13}}{2}} = -\sqrt{\frac{\frac{13}{13} + \frac{12}{13}}{2}} = -\sqrt{\frac{\frac{25}{13}}{2}} = -\sqrt{\frac{25}{26}}. \] Step 2: Simplify \[ \sin \left( \frac{\theta}{2} \right) = -\frac{5 \sqrt{26}}{26}. \]

The correct option is (A) : \(\frac{-5\sqrt{26}}{26}\)

Answer 2

We are given that \(\theta \in (-\pi, 0)\) and \(\cos\theta = -\frac{12}{13}\).

We want to find \(\sin\left(\frac{\theta}{2}\right)\).

Since \(-\pi < \theta < 0\), dividing by 2 gives \(-\frac{\pi}{2} < \frac{\theta}{2} < 0\). This means that \(\frac{\theta}{2}\) is in the fourth quadrant, where sine is negative.

We can use the half-angle formula for sine:

\(\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}\)

Since \(\sin\left(\frac{\theta}{2}\right)\) is negative, we take the negative root:

\(\sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{1 - \cos\theta}{2}}\)

Substituting \(\cos\theta = -\frac{12}{13}\), we have:

\(\sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{1 - \left(-\frac{12}{13}\right)}{2}} = -\sqrt{\frac{1 + \frac{12}{13}}{2}} = -\sqrt{\frac{\frac{25}{13}}{2}} = -\sqrt{\frac{25}{26}}\)

Simplifying the expression:

\(\sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{25}{26}} = -\frac{5}{\sqrt{26}} = -\frac{5\sqrt{26}}{26}\)

Therefore, \(\sin\left(\frac{\theta}{2}\right) = -\frac{5\sqrt{26}}{26}\).