Trig: Write the expression below as a function of twice a given angle. sin 80(degrees)cos 80(degrees)

Question

zepdrix · Answer

Do you remember your double angle formula's for cosine and sine? :)

zepdrix · Answer

$$\sin(2\theta)=2\sin(\theta) \cos(\theta)$$$$\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)$$
Cosine's double angle shows up in a few different forms since it contains squares, but this is the one you'll see most often.

Do you think either of these might help us simplify your expression?

anonymous · Answer

Would I just double both the sin and the cos?

anonymous · Answer

I don't understand trig at all :( I'm trying though

zepdrix · Answer

So we have:$$\sin(80)\cos(80)$$
Hmm that looks very similar to the top equation doesn't it? If it wasn't for that factor of 2?

anonymous · Answer

Yes, so you would double sin(80) then multiply it by cos(80)?

zepdrix · Answer

$$\sin(2*80)=2\sin(80)\cos(80)$$
We want to try and simplify the expression, so it will look something like the thing on the left. Because it ALMOST matches the thing on the right, yes?
Hmm what can we do? :)

anonymous · Answer

Divide 2sin(80)cos(80) by 2?

zepdrix · Answer

Hmm that's a good idea.

zepdrix · Answer

$$\frac{ \sin(2*80) }{ 2 } =\sin(80)\cos(80)$$
Hmm it looks like we successfully simplified our expression down to 1 trig term, which is more convenient sometimes! :)
I know it's a little tricky using all these weird identities, but just take it slow and try to memorize some of these.

anonymous · Answer

Thank you! I have been working on trig identities (online) for a while and am still shaky as you can see, but having someone explain it to me helps a lot!