Questions: Use a half-angle formula to evaluate the expression without using a calculator. csc (15π/8) a. What is the measure of the half-angle α/2 ? α/2= (Type an exact answer in terms of π. Use integers or fractions for any numbers in the expression.)

$Use a half-angle formula to evaluate the expression without using a calculator. csc (15π/8) a. What is the measure of the half-angle α/2 ? α/2= (Type an exact answer in terms of π. Use integers or fractions for any numbers in the expression.)$

Transcript text: Use a half-angle formula to evaluate the expression without using a calculator. \[ \csc \left(\frac{15 \pi}{8}\right) \] a. What is the measure of the half-angle $\frac{\alpha}{2}$ ? \[ \frac{\alpha}{2}=\square \] (Type an exact answer in terms of $\pi$. Use integers or fractions for any numbers in the expression.)

Solution

Solution Steps

To evaluate $\csc \left(\frac{15 \pi}{8}\right)$ using a half-angle formula, we first need to determine the angle $\alpha$ such that $\frac{\alpha}{2} = \frac{15\pi}{8}$. This will allow us to find $\alpha$ and use the half-angle identity for cosecant. The half-angle identity for cosecant is $\csc\left(\frac{\alpha}{2}\right) = \frac{1}{\sin\left(\frac{\alpha}{2}\right)}$, and we can use the sine half-angle formula to find $\sin\left(\frac{\alpha}{2}\right)$.

Step 1: Determine the Half-Angle

We start with the expression $\csc\left(\frac{15\pi}{8}\right)$. To find the measure of the half-angle $\frac{\alpha}{2}$, we set: \[ \frac{\alpha}{2} = \frac{15\pi}{8} \] From this, we can find the full angle $\alpha$ by multiplying both sides by 2: \[ \alpha = 2 \cdot \frac{15\pi}{8} = \frac{15\pi}{4} \]

Step 2: Use the Half-Angle Identity

Next, we apply the half-angle identity for cosecant: \[ \csc\left(\frac{\alpha}{2}\right) = \frac{1}{\sin\left(\frac{\alpha}{2}\right)} \] Using the sine half-angle formula: \[ \sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{2}} \] We need to find $\cos(\alpha)$ where $\alpha = \frac{15\pi}{4}$.

Step 3: Calculate $\cos(\alpha)$

The angle $\frac{15\pi}{4}$ can be simplified by subtracting $4\pi$ (which is equivalent to $8\pi/2$): \[ \frac{15\pi}{4} - 4\pi = \frac{15\pi}{4} - \frac{16\pi}{4} = -\frac{\pi}{4} \] Thus, we have: \[ \cos\left(\frac{15\pi}{4}\right) = \cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Step 4: Substitute $\cos(\alpha)$ into the Sine Formula

Now we substitute $\cos(\alpha)$ back into the sine half-angle formula: \[ \sin\left(\frac{15\pi}{8}\right) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

Step 5: Calculate $\csc\left(\frac{15\pi}{8}\right)$

Finally, we find $\csc\left(\frac{15\pi}{8}\right)$: \[ \csc\left(\frac{15\pi}{8}\right) = \frac{1}{\sin\left(\frac{15\pi}{8}\right)} = \frac{2}{\sqrt{2 - \sqrt{2}}} \]