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.

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.

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

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.

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

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

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

Final Answer