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## IloveCharlie 3 years ago cos67degrees 30'= ?

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1. IloveCharlie

2. ash2326

@IloveCharlie what's 30' in degrees?

3. GOODMAN

Divide 30 seconds by 60. After that, add it to 67. Then take the cosine of it all.

4. IloveCharlie

Hmmm... I got .382. Can you please confirm?

5. ash2326

yeah, it's right but could you use calculator for this? @IloveCharlie

6. IloveCharlie

I just don't know which one it matches up with :/ From the options given

7. IloveCharlie

Whoa @jim_thompson5910 you're typing a lot

8. jim_thompson5910

Yes it's a lot, but it at least gives you the exact answer 67 degrees 30' = 67 degrees + 30/60 = 67+0.5 = 67.5 degress Notice how 2*67.5 = 135 and $\Large \cos(135) = -\frac{\sqrt{2}}{2}$ (using the unit circle) Now turn to the identity $\Large \cos(x) = \sqrt{ \frac{ \cos(2x) + 1 }{2} }$ If we let x = 67.5, then $\Large \cos(x) = \sqrt{ \frac{ \cos(2x) + 1 }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ \cos(2*67.5) + 1 }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ \cos(135) + 1 }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ -\frac{\sqrt{2}}{2} + 1 }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ -\frac{\sqrt{2}}{2} + \frac{2}{2} }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ \frac{-\sqrt{2}+2}{2} }{2} }$ $\Large \cos(67.5) = \sqrt{ \frac{ \frac{2-\sqrt{2}}{2} }{2} }$ $\Large \cos(67.5) = \sqrt{ \left( \frac{2-\sqrt{2}}{2} \right)\left(\frac{1}{2}\right) }$ $\Large \cos(67.5) = \sqrt{ \frac{(2-\sqrt{2})*1}{2*2} }$ $\Large \cos(67.5) = \sqrt{ \frac{2-\sqrt{2}}{4} }$ $\Large \cos(67.5) = \frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}$ $\Large \cos(67.5) = \frac{\sqrt{2-\sqrt{2}}}{2}$

9. jim_thompson5910

sry i had a typo, but i fixed it

10. IloveCharlie

Wow, thanks so much for the awesome step by step explanation! Really helps!

11. jim_thompson5910

you're welcome

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