Question

Given cos 67.5° = [√(2+√2)]/2, find tan 67.5° , simplify where needed, and show work.

I'm starting to learn this stuff, and I'm so confused where to start. I know they gave me the coordinate X as in cos 67.5° = [√(2+√2)]/2, and I also know that tan = sin/cos, but in order for me to get there, I'll need to know sin first right? Please help me.

Answer 1

you know two sides of the triangle..

lets work on that cosine
(2+sqrt2)^2=4+2sqrt2+2=6+2sqrt2

the third side of triangle..
sqrt((2+sqrt2)^2 +2^2)
sqrt(10+2sqrt2)

check that. If correct, then
sine 67.5=(10+2sqrt2)/2

Answer 2

First of all, you have a typo:

<b>cos 67.5 = √(2-√2) /2</b>

Have you come across the identity

(sinØ)^2 + (cosØ)^2 = 1 ?

We can apply it here:

(sin67.5)^2 + (2 - √2)/4 = 1
sin67.5 ^2 = 1 - (2 - √2)/4
= ( 4 - 2 + √2)/4
= (2 + √2)/4
sin67.5 = [√(2+√2)]/2

so tan67.5
= sin67.5/cos67.5
= [√(2+√2)]/2 / [√(2-√2)]/2
= √(2+√2) / √(2-√2)
or
= √[ (2+√2)/(2-√2) ]

take over