Question

Find the exact value by using a half-angle identity. sin seven pi divided by eight

Answer 1

Answer:

$\frac{\sqrt{2-\sqrt{2} } }{2}$

Step-by-step explanation:

Recall the double angle identity for the sine of an angle:

$sin(\frac{\alpha }{2})=+/- \sqrt{\frac{1-cos(\alpha) }{2}$

Therefore, in order to find the $sin(\frac{7\pi}{8} )$, we need to find the cos of twice the angle $\frac{7\pi }{8}$, that is of  $\frac{7\pi }{4}$

We need also to know in which quadrant the $\frac{7\pi }{8}$ angle is, in order to have an idea whether to select the positive or the negative sign given by the formula. We notice that $\frac{7}{8} \pi$ is smaller than $\frac{8}{8} \pi =\pi$. Therefore we are referring to an angle in the upper quadrants, where the sign function is positive. So we are going to go with the positive sign of the double angle formula.

Next, we need to study $cos(\frac{7\pi}{4} )$ to assign a value to our answer:

So we look at $\frac{7}{4} \pi$, as seven times the angle $\frac{\pi}{4}$ (45 degrees), and make a diagram to guide us for the sign and exact numerical value of the cosine function of that angle (see attached picture). We notice that this angle resides in the 4th quadrant, therefore the cos is a positive number. And since we are dealing with one of the well known "special angles" for which we have exact numerical expressions, we can give the answer for $cos(\frac{7\pi}{4} )$:

$cos(\frac{7\pi}{4} )=\frac{\sqrt{2}}{2}$

Now we insert this numerical value into the half angle formula:

$sin(\frac{7\pi }{8})=\sqrt{\frac{1-cos(\frac{7\pi }{4} ) }{2}}=\sqrt{\frac{1-\frac{\sqrt{2}}{2} }{2}}=\sqrt{\frac{\frac{2-\sqrt{2}}{2} }{2}}=\sqrt{\frac{2-\sqrt{2}}{4}}=\frac{\sqrt{2-\sqrt{2} } }{2}$