Question

Given that sin 18 (deg) = (sqrt(5)-1)/4 ... find exact expression for cos 36 (deg).
**cos(2x) = 1-2sin^2x

Answer 1

well 2*18=36

Answer 2

plug into the identity

Answer 3

replace x in cos(2x)=1-2sin^2(x) with 18

Answer 4

I don't get the step where 1-2((sqrt(5)-1)/4)^2 = 1-2((3-sqrt(5))/8)

Answer 5

\[\cos(2 \cdot 18)=\cos(36)=1-2\sin^2(18) =1-2(\sin(18))^2 \\ =1-2 (\frac{\sqrt{5}-1}{4})^2\]

Answer 6

\[=1-2 \frac{(\sqrt{5}-1)^2}{4^2}=1-\frac{2}{16}(\sqrt{5}-1)^2=1-\frac{1}{8}(\sqrt{5}-1)^2\]

Answer 7

multiply the following:
\[(\sqrt{5}-1)^2=(\sqrt{5}-1)(\sqrt{5}-1)=?\]

Answer 8

solution : (sqrt(5) + 1) / 4

Answer 9

that is right

Answer 10

did you get that or is that the answer in the back?

Answer 11

that's what it says in the back..

Answer 12

ok do you still want help on this...

Answer 13

I was trying to get you to multiply (Sqrt(5)-1)(sqrt(5)-1)

Answer 14

yes

Answer 15

\[(\sqrt{5}-1)^2 \\ =(\sqrt{5}-1)(\sqrt{5}-1) \\ =\sqrt{5}(\sqrt{5}-1)-1(\sqrt{5}-1)=?\]

Answer 16

\[6 - \sqrt{5}-\sqrt{5}\]

Answer 17

\[(\sqrt{5}-1)^2=5-\sqrt{5}-\sqrt{5}+1=6-2 \sqrt{5}\]
good so we have
\[\cos(2 \cdot 18)=\cos(36)=1-2\sin^2(18) =1-2(\sin(18))^2 \\ =1-2 (\frac{\sqrt{5}-1}{4})^2 =1-2(\frac{6- 2 \sqrt{5}}{16})=1-\frac{6-2 \sqrt{5}}{8} \\ =\frac{8}{8}-\frac{6-2 \sqrt{5}}{8}=\frac{8-(6-2 \sqrt{5})}{8}\]

Answer 18

simplify the top

Answer 19

\[8-(6-2 \sqrt{5})=8-6+2\sqrt{5}=?\]

Answer 20

4\[4\sqrt{5}\]

Answer 21

well 8-6 is 2
so \[8-(6-2 \sqrt{5})=8-6+2\sqrt{5}=2+2 \sqrt{5}\]

Answer 22

you can't do anything else with the top except factor a 2 out like so
\[2(1+\sqrt{5})\]

Answer 23

so we have \[\cos(2 \cdot 18)=\cos(36)=1-2\sin^2(18) =1-2(\sin(18))^2 \\ =1-2 (\frac{\sqrt{5}-1}{4})^2 =1-2(\frac{6- 2 \sqrt{5}}{16})=1-\frac{6-2 \sqrt{5}}{8} \\ =\frac{8}{8}-\frac{6-2 \sqrt{5}}{8}=\frac{8-(6-2 \sqrt{5})}{8} \\ =\frac{2(1+\sqrt{5})}{8}\]

Answer 24

and then divide top and bottom by 2

Answer 25

and we know 2/8 reduces to?

Answer 26

yea

Answer 27

so you would have what you seek

Answer 28

would ya look at that.

Answer 29

who woulda thunk it

Answer 30

a little bit of algebra
but does everything I asked you to do and everything I did make sense?

Answer 31

do you have any question about what I did?

Answer 32

no i got it <3

Answer 33

thanks

Answer 34

cool