Use a half-angle identity to find the exact value of this expression. 

Given sin = , 0° <  < 90°,

find cos 

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We are given sin = 

cos = 

 is a positive number, the sine is positive in the first two
 quadrants so 0° <  < 180° and 0° <  < 90°,

so we take the positive and cos = 

So everything's positive and in the first quadrant.

Since sin =  and the sine is the opposite over the hypotenuse,
we draw a right triangle containing  with the length of the opposite side as
the numerator of  and the hypotenuse as the denominator of . 

Before drawing the right triangle, we calculate the adjacent side using the
Pythagorean theorem:

c² = a² + b²
3² = a² + 
 9 = a² + 4·2
 9 = a² + 8
 1 = a²
 1 = a 

So the right triangle is like this: 



cos = 

From the right triangle, we can get cos by
using the fact that the cosine is the adjacent over the 
hypotenuse .  Substituting:

cos = 

To simplify the compound fraction under the radical we multiply
top and bottom by 3

cos = 

cos = 

cos = 

cos = 

Rationalize the denominator by multiplying top and bottom by 3

cos = 

cos = 

cos = 

Edwin