Question

What is the value of cos 17 cos 81 + sin 43 cos 69 + 1/2 cos 142?

Answer 1

Could you show us what you've done so far please?

Answer 2

Yes, it is the original question :-(
Maybe you can calculate cos 17 or cos 81 or sin 43 or cos 69 or cos 142?

Answer 3

IF not, you can just use a calculator and find the value to the nearest 2 decimal places or something.

Answer 4

But I'm guessing RolyPoly is right about needing the exact value.

Answer 5

Yes, but without calculator

Answer 6

Well first I can see that sin43 can be rewritten as cos(90-43)

Answer 7

Then cos(90-43)=cos(47)
Which can be rewritten again as cos(30+17)

Answer 8

Then you use angle results.

Answer 9

ANd then cos81 can be rewritten as cos(150-69) and you can use angle results on that. And you know that:
\[\cos150=-\frac{\sqrt{3}}{2}\]
and:
\[\sin150=\frac{1}{2}\]

Answer 10

@RolyPoly I agree that would work as well, but you would just form another different angle which is 21, so it would be a bother to find that as well.

Answer 11

Then?

Answer 12

So your equation becomes this:
\[\cos17[\cos(150-69) + \cos(30+17)\cos69 + \frac{1}{2}\cos142=?\]
COuld you expand the double angle results please.

Answer 13

Everything will cancel, when you expand everything out the way I planned it for you.

Answer 14

And then you would be left with:
\[\frac{1}{2}\cos142\] What I need you to do is show me that you understand YOUR OWN question.

Answer 15

You don't just let me do the talking. You need to really do it yourself. I think I've done 3/4 of it. You should be capable of going on. I will be here to guide you from now on. So you should show me what you've done from here on then.

Answer 16

My answer is still at
\[\frac{ 1 }{ 2 } (\sin 69 \cos 17 + \cos 69 \sin 17) + \frac{ 1 }{ 2 } \cos 142\]
\[\frac{ 1 }{ 2 } (\sin 86 + \cos 142)\]
Then what should i do?

Answer 17

@RolyPoly and @fiedar
I will scan the working out for you UP to this stage and you can see everything cancels out.

Answer 18

Ok, thank you. I will wait for it

Answer 19

It looks like you've gone ahead with it. Yeah you've gotten it right. Just a mistake at your second last line. It's meant to be a minus sign inside those brackets/paranthesis.

Answer 20

And then what do you think we should do?

Answer 21

Oh, I see. it must be negative. Then?

Answer 22

Nah it's just a minor mistake. You're fine with the last line. Just the second last line.

Answer 23

So when you have sin86 it can be written as sin(60+26). ANd then I can see something happening with cos142. It can be written as cos(90+52). ANd the angle 52 is twice that of 26. So it would give you an idea of what to do now.

Answer 24

YOu would have to expand cos142 twice due to the double angle of 52.

Answer 25

Oh, OK. Thank you for your help :D

Answer 26

Now worries man, I will put up my scanned working out if you need to look on something.

Answer 27

Wait something's wrong.

Answer 28

Sorry you got the second line and last line wrong.

Answer 29

I was right about the minus sing but the last line should be 1/2(sin52 +cos142)

Answer 30

minus sign*

Answer 31

so you wouldn't need to do much with the double angle results.

Answer 32

Yes. Thank you once more

Answer 33

And you're right @RolyPoly Both methods are correct. Well Done mate.