SOLUTION: Im a triangle ABC. Cos A/a = Cos B/b = Cos C/c, and a = 2. It is possible to compute the area of the triangle, (Hint: compare the problem data with the Law of the Sines, compute

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Question 1199591: Im a triangle ABC.
Cos A/a = Cos B/b = Cos C/c, and a = 2. It is possible to compute the area of the
triangle, (Hint: compare the problem data with the Law of the Sines, compute ratios between the sines and cosines of the angles)
A) sqrt3/4
B) sqrt3/2
C) sqrt3
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
I'm a triangle ABC.
Cos A/a = Cos B/b = Cos C/c, and a = 2. It is possible to compute the area of the
triangle, (Hint: compare the problem data with the Law of the Sines, compute ratios between the sines and cosines of the angles)
A) sqrt3/4
B) sqrt3/2
C) sqrt3
~~~~~~~~~~~~~~~~~~~ 

We are given  Cos(A)/a = Cos(B)/b = Cos(C)/c.    (1)


Let "p" be the common value of all these three ratios

     =  =  = p.    (2)

Then

    cos(A) = pa,    (3)

    cos(B) = pb,    (4)

    cos(C) = pc.    (5)


From the other side, we have this "Law of Sines" formula

     =  = .    (6)


Let "q" be the common value of all these three ratios

     =  =  = q.    (7)

Then

    sin(A) = qa,    (8)

    sin(B) = qb,    (9)

    sin(C) = qc.    (10)


Divide (8) by (3);  divide (9) by (4);  divide (10) by (5).  You will get then

     = ,

     = ,

     = ,


or  tan(A) = tan(B) = tan(C) = .


The tangents of three angles of a triangle are equal --- hence the angles are congruent A = B = C.


So, the triangle ABC is an equilateral triangle with the side length of 2 units.


Hence, its area is   =  =  square units.

Solved.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

cos(A)/a = cos(B)/b = cos(C)/c
cos(A)/a = cos(B)/b
cos(A)/cos(B) = a/b

Use the law of sines to apply similar steps:
sin(A)/a = sin(B)/b = sin(C)/c
sin(A)/a = sin(B)/b
sin(A)/sin(B) = a/b

We can see that:
cos(A)/cos(B) = a/b
cos(A)/cos(B) = sin(A)/sin(B)
which can be rearranged into
sin(B)/cos(B) = sin(A)/cos(A)
tan(B) = tan(A)
B = A
A = B

Follow a similar chain of logic to conclude that B = C.

We have A = B and B = C, so A = C
All three angles are the same measure.
A+B+C = 180
A+A+A = 180
3A = 180
A = 180/3
A = 60
Each angle is 60 degrees.
The triangle is equilateral.

Area of equilateral triangle with side length 's'
Area = 0.25*s^2*sqrt(3)
Area = 0.25*2^2*sqrt(3)
Area = sqrt(3)


Answer: C) sqrt(3)
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