## IloveCharlie Group Title cos67degrees 30'= ? 2 years ago 2 years ago

1. IloveCharlie Group Title

2. ash2326 Group Title

@IloveCharlie what's 30' in degrees?

3. GOODMAN Group Title

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

4. IloveCharlie Group Title

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

5. ash2326 Group Title

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

6. IloveCharlie Group Title

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

7. IloveCharlie Group Title

Whoa @jim_thompson5910 you're typing a lot

8. jim_thompson5910 Group Title

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 Group Title

sry i had a typo, but i fixed it

10. IloveCharlie Group Title

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

11. jim_thompson5910 Group Title

you're welcome