Question

cos67degrees 30'= ?

Answer 1

@IloveCharlie what's 30' in degrees?

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

Whoa @jim_thompson5910 you're typing a lot

Answer 7

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

\[\Large \cos(x) = \sqrt{ \frac{ \cos(2x) + 1 }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ \cos(2*67.5) + 1 }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ \cos(135) + 1 }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ -\frac{\sqrt{2}}{2} + 1 }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ -\frac{\sqrt{2}}{2} + \frac{2}{2} }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ \frac{-\sqrt{2}+2}{2} }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{ \frac{2-\sqrt{2}}{2} }{2} }\]

\[\Large \cos(67.5) = \sqrt{ \left( \frac{2-\sqrt{2}}{2} \right)\left(\frac{1}{2}\right) }\]

\[\Large \cos(67.5) = \sqrt{ \frac{(2-\sqrt{2})*1}{2*2} }\]

\[\Large \cos(67.5) = \sqrt{ \frac{2-\sqrt{2}}{4} }\]

\[\Large \cos(67.5) = \frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}\]

\[\Large \cos(67.5) = \frac{\sqrt{2-\sqrt{2}}}{2}\]

Answer 8

sry i had a typo, but i fixed it

Answer 9

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

Answer 10

you're welcome