sin(x+a)=sqrt(2)cos(x-a): find tan x

By Anonymous on Wednesday, May 31, 2000 - 01:35 am :

I haven't got a clue how to prove the following question. Can anyone help?

Given that sin(x+a) =sqrt(2)cos(x-a), where cosx x cosa is not equal to 0,

a] prove that tanx = sqrt(2)-tan(a)/(1-sqrt(2)tan(a))

Thanks

By Harry Smith (Harry) on Wednesday, May 31, 2000 - 11:53 am :

You will need to use your compound angle formulae to do this question. If you see something like cos(x +a) it's always a good idea to see if you can use any identities you know already. Using 2 instead of root 2 (it makes no difference), your original equation was:

sin(x +a)=2cos(x -a)

Expanding this out using compound angle formulae gives you:

sinx cosa+ cosx sina = 2cosx cosa+ 2sinx sina

Now we can try and find a clue about what to do next from the original question. The question states that cosx cosa is not equal to 0. This suggests that you will need to divide by cosx cosa at some point. Do this to both sides of the equation, remembering that tanx = sinx /cosx, and you have:

tanx + tana = 2 + 2tanx tana.

You are now essentially done. Collect the terms with tanx on one side, factor our tanx, and divide both sides by the other factor.

You now have:

tanx = (2 - tana)/(1 - 2tana)

as required.

Hope this is clear.

Harry

By Anonymous on Wednesday, May 31, 2000 - 12:16 pm :

Harry,

That was very clear, thanks very much.

I was able to expand the compound angle formula, but didn't know what to do next. I can see the usefulness of hints such as the product not being equal to 0. I'll remember that. It also made common sense to use 2 instead of the squareroot of 2! I'll remember that too.

Thanks again Harry