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sinC = (sinA + sinB) / (cosA + cosB), shape of ABC = ?

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Given triangle ABC.

sinC = (sinA + sinB) / (cosA + cosB)

What details/characteristics can be determined about the shape of ABC (e.g. type of triangle) ?

Mar 29, 2019

#1
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$$\textrm{Let}\quad\overline{BC}=a,\quad\overline{CA}=b,\quad\overline{AB}=c\\ \frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R\quad (\textrm{R=radius of the circumscribed circle of triangle ABC})\\ \therefore sinA=\frac{a}{2R},\quad sinB=\frac{b}{2R},\quad sinC=\frac{c}{2R}\\ cosA=\frac{b^2+c^2-a^2}{2bc},\quad cosB=\frac{c^2+a^2-b^2}{2ca},\quad cosC=\frac{a^2+b^2-c^2}{2ab}\\ \therefore \frac{c}{2R}=\frac{\frac{a}{2R}+\frac{b}{2R}}{\frac{b^2+c^2-a^2}{2bc}+\frac{c^2+a^2-b^2}{2ca}}\\ \quad c=\frac{2abc(a+b)}{a(b^2+c^2-a^2)+b(c^2+a^2-b^2)}=\frac{2abc(a+b)}{ab^2+c^2a-a^3+bc^2+a^2b-b^3}=\frac{2abc(a+b)}{c^2(a+b)-(a-b)^2(a+b)}=\frac{2abc}{c^2-(a-b)^2}\\ \therefore c^2-(a-b)^2=2ab\\ \quad c^2=2ab+(a-b)^2=a^2+b^2\\ \therefore \textrm{Triangle ABC is a right triangle that has angle C as the right angle.}$$

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Mar 30, 2019
#2
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Hi ThorganHarzard

Welcome to the   Web2.0calc   forum

That is a very impressive answer but you lost me at the ''=2R"   bit.    What has it to do with the circumscribed circle?

I am not questioning your knowledge, I just don't understand.

Melody  Mar 30, 2019
edited by Melody  Mar 30, 2019
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The 2R comes from the standard proof of the sine rule.

Draw your triangle ABC with its circumscribing circle, radius R, centre O.

With the usual notation, angle BOC = 2A, so if you drop a perpendicular from O to BC you get a rt-angled triangle with

two sides R and a/2 and an angle A, from which a/2R = sin(A) and a/sin(A) = 2R.

Repeat for B and C and equate the three.

Here's an alternative answer to the original question.

LHS,

$$\displaystyle \sin(C)=2\sin(C/2)\cos(C/2)=2\sin(90-(A+B)/2)\cos(90-(A+B)/2)\\ =2\cos((A+B)/2)\sin((A+B)/2) \dots\dots\dots(1)$$

RHS

$$\displaystyle \frac{\sin(A)+\sin(B)}{\cos(A)+\cos(B)}=\frac{2\sin((A+B)/2)\cos((A-B)/2)}{2\cos((A+B)/2)\cos((A-B)/2)}=\frac{\sin((A+B)/2)}{\cos((A+B)/2)}\dots\dots(2)$$

Equate (1) and (2), cancel and cross multiply,

$$\displaystyle \cos^{2}((A+B)/2)=1/2,\\\text{so }\cos((A+B)/2) = 1/\sqrt{2}\\\text{so }(A+B)/2=45\text{ deg}\\A+B=90\text{ deg}\\C=90\text{ deg}.$$

Tiggsy.

Mar 31, 2019
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Thanks Tiggsy.

Melody  Mar 31, 2019
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Welcome as a member Tiggsy.

I should have said this immediately but I did not realize straight away that you had joined!

I know you have been here forever but I finally talked you into becoming a member.  I am very pleased!

Now I will be able to look back and find your fabulous answers (from this point forward)

Finally ... I have exhausted this question.  There was a lot for me to work through here.

I do not remember seeing those formulas before that you used Tiggsy. Hopefully now they, and their proofs, are a little embedded into my brain.

Anyway, i finally fully understand both solutions so I am happy.

Melody  Apr 11, 2019