Questions: Use the given information to find the exact function value. Given cos α = 3/5 and 3π/2 < α < 2π, find sin α/2.

Transcript text: Use the given information to find the exact function value. Given $\cos \alpha=\frac{3}{5}$ and $\frac{3 \pi}{2}<\alpha<2 \pi$, find $\sin \frac{\alpha}{2}$.

Solution

Solution Steps

To find $\sin \frac{\alpha}{2}$, we can use the half-angle identity for sine. The half-angle identity for sine is given by:

\[ \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}} \]

Since $\frac{3 \pi}{2} < \alpha < 2 \pi$, $\alpha$ is in the fourth quadrant, where cosine is positive and sine is negative. Therefore, $\frac{\alpha}{2}$ will be in the second quadrant, where sine is positive. We will use the positive square root.

Solution Approach

Use the half-angle identity for sine.
Substitute the given value of $\cos \alpha$ into the identity.
Calculate the value.

Step 1: Given Information

We are given that $ \cos \alpha = \frac{3}{5} $ and that $ \frac{3\pi}{2} < \alpha < 2\pi $. This indicates that $ \alpha $ is in the fourth quadrant.

Step 2: Half-Angle Identity

To find $ \sin \frac{\alpha}{2} $, we use the half-angle identity: \[ \sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}} \]

Step 3: Substitute the Value

Substituting $ \cos \alpha = \frac{3}{5} $ into the half-angle identity: \[ \sin \frac{\alpha}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{\frac{2}{5}}{2}} = \sqrt{\frac{2}{10}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]

Step 4: Calculate the Value

Calculating the numerical value: \[ \sin \frac{\alpha}{2} \approx 0.4472 \]