Questions: If cos A = 7/8 with A in QIV, find sin (A/2). sin (A/2) = -sqrt(3)/5 sin (A/2) = 1/5 sin (A/2) = sqrt(3)/5 sin (A/2) = -1/5 sin (A/2) = 1/4

Transcript text: If $\cos A=\frac{7}{8}$ with A in QIV, find $\sin \left(\frac{A}{2}\right)$. $\sin \left(\frac{A}{2}\right)=-\frac{\sqrt{3}}{5}$ $\sin \left(\frac{A}{2}\right)=\frac{1}{5}$ $\sin \left(\frac{A}{2}\right)=\frac{\sqrt{3}}{5}$ $\sin \left(\frac{A}{2}\right)=-\frac{1}{5}$ $\sin \left(\frac{A}{2}\right)=\frac{1}{4}$

Solution

Solution Steps

To find $\sin \left(\frac{A}{2}\right)$ given $\cos A = \frac{7}{8}$ with $A$ in the fourth quadrant, we can use the half-angle identity for sine: \[ \sin \left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{2}} \] Since $A$ is in the fourth quadrant, $\frac{A}{2}$ will be in the second quadrant, where sine is positive. Therefore, we will use the positive root.

Step 1: Given Information

We are given that $ \cos A = \frac{7}{8} $ and that angle $ A $ is in the fourth quadrant.

Step 2: Half-Angle Identity

To find $ \sin \left(\frac{A}{2}\right) $, we use the half-angle identity: \[ \sin \left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{2}} \]

Step 3: Substitute the Value of $ \cos A $

Substituting $ \cos A = \frac{7}{8} $ into the half-angle identity: \[ \sin \left(\frac{A}{2}\right) = \sqrt{\frac{1 - \frac{7}{8}}{2}} = \sqrt{\frac{\frac{1}{8}}{2}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

Step 4: Determine the Sign

Since $ A $ is in the fourth quadrant, $ \frac{A}{2} $ will be in the second quadrant where sine is positive. Therefore, we take the positive root.