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Why approximate? I will show you what you should know being in the trig section. Law of cosines. Degree mode!!

a = 4 (angle opposite = alpha)

b = 5 ( angle opposite = beta)

c = 8 ( angle opposite = gamma )

a^2 = b^2 + c^2 - 2bc cos(alpha)

4^2 = 5^2 + 8^2 - 2(5)(8) cos(alpha)

16 = 89 - 80 cos(alpha)

-73 = -80 cos(alpha)

0.9125 = cos(alpha)

arcos(0.9125) = alpha

alpha = 24.15 degrees

------------------------------

b^2 = a^2 + c^2 - 2bc cos(beta)

5^2 = 4^2 + 8^2 - 2(4)(8) cos(beta)

25 = 80 - 64 cos(beta)

-55 = -64 cos(beta)

0.859375 = cos(beta)

arcos(0.859375) = beta

beta = 30.75 degrees

---------------------------------

Now to find gamma, subtract from 180 degrees

180 - 24.15 - 30.75

= 125.1 degrees

alpha = 24.15 degrees ( subject to rounding, but all add to 180 degrees )

beta = 30.75 degrees

gamma = 125.1 degrees

now you see the smallest, the angle opposite the a side, which is 4

( be in degree mode!!)

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Q: What is the approximate size of the smallest angle of a triangle whose sides are 4 5 and 8?
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