cosine and sine

**magentarita** · May 5th 2009, 09:17 PM

I will use d for degrees.

cos80dcos20d - sin80dsin20d =

I got -0.173648178

Is this correct?

**Grandad** · May 5th 2009, 10:04 PM

Hello magentarita

Originally Posted by magentarita

I will use d for degrees.

cos80dcos20d - sin80dsin20d =

I got -0.173648178

Is this correct?

You are quite correct!

(By the way, did you spot that you can use $\cos(A+B)=\cos A\cos B-\sin A \sin B$ , and say that $\cos 80^o\cos 20^o-\sin 80^o \sin 20^o=\cos 100^o?$ )

Grandad

**magentarita** · May 6th 2009, 05:37 AM

Originally Posted by Grandad

Hello magentaritaYou are quite correct!

(By the way, did you spot that you can use $\cos(A+B)=\cos A\cos B-\sin A \sin B$ , and say that $\cos 80^o\cos 20^o-\sin 80^o \sin 20^o=\cos 100^o?$ )

Grandad

So, cos100 degrees is the same as -0.173648178 but how I find cos100 degrees using the given formula?

**Grandad** · May 6th 2009, 09:34 AM

Hello magentarita

Originally Posted by magentarita

So, cos100 degrees is the same as -0.173648178 but how I find cos100 degrees using the given formula?

If $A = 80$ and $B = 20$ , then the formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$ becomes

$\cos(80+20)=\cos100=\cos80\cos20-\sin80\sin20$

So to work out the value of $\cos80\cos20-\sin80\sin20$ you can simply use a calculator and find $\cos100$ instead.

Or, of course, you can do it the long way and find the separate values of $\cos80, \cos20, \sin80$ and $\sin20$ , and then multiply and add. You should get the same answer whichever way you choose.

Grandad

**magentarita** · May 6th 2009, 01:58 PM

Originally Posted by Grandad

Hello magentaritaIf $A = 80$ and $B = 20$ , then the formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$ becomes

$\cos(80+20)=\cos100=\cos80\cos20-\sin80\sin20$

So to work out the value of $\cos80\cos20-\sin80\sin20$ you can simply use a calculator and find $\cos100$ instead.

Or, of course, you can do it the long way and find the separate values of $\cos80, \cos20, \sin80$ and $\sin20$ , and then multiply and add. You should get the same answer whichever way you choose.

Grandad

I need to find two numbers that when added together produce 100. I wanted to learn how to find the answer without a calculator. Can you show me?

**Grandad** · May 6th 2009, 09:53 PM

Hello magentarita

Originally Posted by magentarita

I need to find two numbers that when added together produce 100. I wanted to learn how to find the answer without a calculator. Can you show me?

I'm not quite sure what you mean by this. Obviously you don't mean that you can't find two numbers that add together to make 100.

Do you mean how do you find $\cos100^o$ without using a calculator? If so, I'm not sure that you can. There are some angles - like $30^o, 45^o, 60^o, 90^o$ and so on - that have easy sines and cosines, and you may not need a calculator for these. And you may be able to find two of these special angles that will combine to give $100^o$ (perhaps by adding, subtracting, doubling, and so on), but I can't think of any at present.

Is this what you meant?

Grandad

**magentarita** · May 7th 2009, 06:14 AM

Originally Posted by Grandad

Hello magentaritaI'm not quite sure what you mean by this. Obviously you don't mean that you can't find two numbers that add together to make 100.

Do you mean how do you find $\cos100^o$ without using a calculator? If so, I'm not sure that you can. There are some angles - like $30^o, 45^o, 60^o, 90^o$ and so on - that have easy sines and cosines, and you may not need a calculator for these. And you may be able to find two of these special angles that will combine to give $100^o$ (perhaps by adding, subtracting, doubling, and so on), but I can't think of any at present.

Is this what you meant?

Grandad

Yes, this is exactly what I mean.

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