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Category: Pre-Calculus
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Experience:  Taught 40 years College & HS * MS in Math * Published Writer
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# find the exact value of cos 67.5 degrees using the half angle

### Customer Question

find the exact value of cos 67.5 degrees using the half angle identity.
Submitted: 5 years ago.
Category: Pre-Calculus
Expert:  mijcar replied 5 years ago.

mijcar :

I will help with this.

Customer:

ok

mijcar :

We know that the cos (135 degrees) = -sqrt(2)/2.

mijcar :

The half angle formula states that cos (x/2) = +/- sqrt [ (1 - cos s)/2 ]

mijcar :

So cos (67.5 degrees) = cos (135 degrees / 2)

Customer:

it says simplify the answer including any radicals. Use integers or fractions for any numbers in the expression. rationalize all denominators.

mijcar :

= +/- sqrt [ (1 -cos(135 deg) ) / 2 ]

mijcar :

I will do that.

mijcar :

= +/- sqrt [ (1 - (-sqrt(2)/2 ) /2 }

mijcar :

= +/- sqrt [ (1 + sqrt(2)/2 ) /2]

mijcar :

= +/- sqrt [ (2 + sqrt(2) ) 2 ]

mijcar :

I meant: = +/- sqrt [ (2 + sqrt(2) ) / 2]

Customer:

so 2 + sqrt 2 /2

mijcar :

hold on a minute. It's hard to input into the chat.

Customer:

ok

mijcar :

I keep hitting the enter key too fast. Let me write that last line carefully.

Customer:

ok

mijcar :

= +/- sqrt [ ( (2 + sqrt(2) )/2 ) /2 ]

mijcar :

there.

mijcar :

= +/- sqrt [ (2 + sqrt(2) ) /4 ]

mijcar :

= +/- sqrt (2 + sqrt (2) ) / 2

Customer:

so 2 + sqrt 2 over 4

Customer:

mijcar :

The 4 is in the denominator of the square root. So it "comes out of the square root as a 2.

Customer:

so is the answer 2 + sqrt 2 over 2?

mijcar :

Remember the parentheses and the sqrt on top. This solution is a square root with another square root inside it. I will show it in a moment.

mijcar :

sqrt ( 2 + sqrt 2 )

mijcar :

2

Customer:

it said that was wrong again

mijcar :

How are you writing it?

Customer:

sqrt (2+sqrt2)/2

mijcar :

try: (sqrt(2 + sqrt(2))/2

Customer:

no its not excepting that.

mijcar :

I apologize for typing this too fast. It should be

mijcar :

(sqrt(2-sqrt(2) ) / 2

Customer:

mijcar :

yes

Customer:

ok thanks im not sure but i will except it.