Using the law of cosines, write an algebraic proof to show that the angles in an equilateral triangle must equal 60°. Use "∧" to indicate exponents. For example, type a2 as a∧2.
(Hint: Let s be a length of each side and x be the angle measure; then use these variables in the law of cosines.)

Question

Using the law of cosines, write an algebraic proof to show that the angles in an equilateral triangle must equal 60°. Use "∧" to indicate exponents. For example, type a2 as a∧2.
(Hint: Let s be a length of each side and x be the angle measure; then use these variables in the law of cosines.)

radar · Answer

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radar · Answer

Personally I would use the fact that the sum of the three angles of a triangle equal 180 degrees (3x = 180) thus x = 60 degrees.  But alas, we are told to use the Law of Cosines.

anonymous · Answer

so using the law of cosines how would that work? @radar

radar · Answer

Law of Cosines, "c^2 = a^2 + b^2 - 2abCos c
In this case a = s, b = s, and c=s
substituting in the equation we now have
s^2 = s^2 + s^2 - 2 ssCosx
s^2 = 2s^2 -2s^2Cos x
-2s^2Cosx=s^2 - 2s^2
-2s^2Cosx = -s^2
Cosx =-s^2/-2s^2 = 1/2 = .5
Leaving the last step as an exercise for @Valerie709

radar · Answer

Just take the inverse Cos of .5 to get the angle.

anonymous · Answer

I just plug that in my calculator using the cos-1 button right? @radar

radar · Answer

Yes, on mine I have to use the shift button, then the Cos button.

radar · Answer

$$Cos ^{-1}$$

radar · Answer

What did you get?

anonymous · Answer

and that's it nothing else? the problem is complete with that? @radar

radar · Answer

I would think so, using the law of cosines with a triangle that has 3 equal sides, the angles will be 60 degrees, we only did it once, but doing it for each angle x would appear identical.

radar · Answer

What did you get for the value of x in degrees???

anonymous · Answer

okay can you help me with this one please? @radar 

In a  ABC, if  B = 60° and the ratio of two sides is a : c = 2 : √3 + 1, then  A= 45°.
True or False?

radar · Answer

Were there instructions on the procedures or what law is used to solve this?

anonymous · Answer

no. @radar but my lesson is tittled law of cosines so I'm assuming law of cosines?

radar · Answer

Probably a good assumption.  O.K. let me ponder this to see how to employ the law of cosines to verify that angle A is 45 degree.....

anonymous · Answer

okay thank you! @radar

radar · Answer

I haven't figured out how to use that a/c ratio to determine angle A.  And it is pass my bedtime here in the Ozarks of MO.  Good luck on this, and suggest you repost it as a new problem.

anonymous · Answer

alright well thank you so much for even giving it a try! @radar