# Trig substitution

1. Jun 16, 2009

### DERRAN

1. The problem statement, all variables and given/known data
If sin54=P, express the following in terms of p, with out using a calculator.
1] cos18
2] $$\frac{tan27+cot63}{1+tan207.cot117}$$

2. Relevant equations

3. The attempt at a solution

1] cos18=sin(90-18)
=sin(72)

sin72=sin(180-108)
=sin(180-108)
=sin180.cos108-cos180.sin108
=0-(-sin108)
=sin108

sin108=2.sin54.cos54
=2pcos54.$$\sqrt{1-p^{2}}$$

2] =$$\frac{2tan27}{1-tan27.tan27}$$

=$$\frac{2tan27}{1-tan^{2}27}$$

=tan2(27)

=tan54

=$$\frac{p}{\sqrt{1-p^{2}}}$$

2. Jun 16, 2009

### HallsofIvy

"P" or "p"? You shouldn't use both to mean the same thing.

Well, you mean, of course,
$$sin(108)= 2p\sqrt{1- p^2}$$
Also you haven't finished the problem. Yes, cos(18) is equal to that but you should write that explicitely:
$$cos(18)= 2p\sqrt{1- p^2}$$
I see nothing wrong with (2).

3. Jun 16, 2009

### DERRAN

Thanks, I made a few typing errors.