Find the A, B and C angles in a triangle in which the following equation holds.?

Question

cosA cosB + sinA sinB sinC=1

Please show the details.

Answer 1

Purely by accident, I realise that it holds for a
45-45-90 triangle
A = B = 45
C = 90

You can recognize that if sinC = 1 and A = B
then you end up with cos^2 A + sin^2 A = 1

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Here's how you can prove it formally.
sinC = sin(A+B)

Let f(A,B) = cosA*cosB + sinA*sinB*sin(A+B)
f is a function of two independent variables, A and B
We wish to prove that the maximum value of f = 1

Take partial derivatives, fA and fB
fA = -sinA*cosB + cosA*sinB*sin(A+B) + sinA*cosB*cos(A+B)
fB = -cosA*sinB + sinA*cosB*sin(A+B) + sinA*cosB*cos(A+B)

We require fA = 0 = fB
Subtracting the two equations and simplifying:
sin(A-B)*[1 + sin(A+B)] = 0
So either A=B or A+B = 90

If A+B = 90,
fA, fB = 0 implies sin(A-B) = 0 or A=B

If A=B, sin(2A) + cos(2A) = 1
thus A = 45 = B

Only solution where fB = fA = 0 is
A = B = 45

f(45, 45) = 1
Thus the only solution is A = B = 45, C = 90

Answer 2

cosA cosB + sinA sinB sinC=1
Taking chances
1 / 2 + 1 / 2 = 1

cosA cosB = 1 / 2

1 / √2 . 1 / √2 = 1 / 2
cos45 cos45 = 1 / 2
That also means

sin45sin45sinC = 1 / 2
1 / 2 sinC = 1 / 2
sinC = 1
sin90 = 1
C = 90
A = 45, B = 45, C = 90

Answer 3

By inspection

A = 45°
B = 45°
C = 90°

cosA cosB + sinA sinB sinC

= cos45° cos45° + sin45° sin45° sin90°

= (1/√2)(1/√2) + (1/√2)(1/√2)(1) = 1/2 + 1/2 = 1

Answer 4

cosA cosB + sinA sinB sinC = a million if this holds, then suitable angled (say C=ninety) and isosceles cosA cosB + sinA sinB *sin ninety = a million cos (A-B) =a million = cos 0 = cos 2pi A=B or ----(a million) A= B + 2pi ---(2) A+B+C=a hundred and eighty A+B= a hundred and eighty-ninety =ninety 2A=2B =ninety A=B=40 5 --------------------------------- sine rule provides sinA/a = sin B/b = sin C/c sin A/a = sin A/b = sin ninety/c a=b=c it says that trianle is likewise isosceles ------------------------ it is real with (2) additionally sin B/b = sin (2pi+A)/b = sin ninety/c sin A/a = sin A/b = sin ninety/c a=b=c it says that trianle is likewise isosceles