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Is there only one way to evaluate $\cos \left(\frac{5 \pi}{4}\right)$ ? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

$\cos \left(\frac{5 \pi}{4}\right)$

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Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Alayna H.

McMaster University

Kayleah T.

Harvey Mudd College

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Video Transcript

Okay, let's solve this problem in two different ways. What is co sign of five pi divided by four? Well, we can split it up as for example, co sign of three pi over four plus two pi over four. And that's just the same as co sign of three pi over four plus pie or two. Now we can use our coastline addition rule which we can use to write this same expression as co sign of three pi over four times co sign of pirate too minus the sign of three pi over four timesthe sign of pie or two and then we can evaluate all these terms. This is gonna be negative radical to over two. This is going to be zero. This is going to be radical to over two, and this is gonna be one. So that simplifies to negative radical to over two. Let's do it a second way. We can also split up co sign of five pi divided by four as co sign of four pi divided by four plus pi divided by four or simplified pi plus pi over four And so using the same exact rule co sign of pi times Go sign of pied Abide by four minus Sign of pie Sign Pi divided by four and then we evaluate again becomes negative One becomes a radical to or two This becomes zero and this is radical to or two. So it simplifies to negative radical to over two, which is exactly what we got before. And so there are multiple ways.