Law Of Sines, deducing addition formula

[xavierhart]

I'm attempting to self [URL='https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/']study mathematics[/URL]. Will be starting college next year and recently got a good score on the SAT Math 2 (~700). I bought "Geometry Revisited" by Coxeter and Greitzer, hoping to gain a bit more knowledge of Geometry. I got stuck on the first question! Until now, all the knowledge I have of geometry was from preparing for my SAT test.

This was the first exercise in the book and so far it has only stated the Law of Sines, so I'm assuming that I won't need anything else to show that a=b cos C + c cos B. Especially seeing at it says that in the question! Haha.

Homework Statement

Show that, for any triangle ABC, even if B or C is an obtuse angle, a=b cos C + c cos B. Use the Law of Sines to deduce the "addition formula":

sin(B+C) = sin B cos C + sin C cos B

Homework Equations

Law of Sines is all that has been stated so far.

The Attempt at a Solution

Non-existent as of yet, apart from stating different sides in terms of trig functions of the angles, but getting nowhere.

I understand this question may be quite trivial in the end seeing as it is the first one in the book, but I would appreciate any points in the right direction.

 

[xavierhart]

So I worked out a=b cos C + c cos B. Because the altitude to BC divides a into b cos C and c cos B.. Makes perfect sense now that I look at it.

Although, still having trouble getting to the addition formula. I've gone through the proof of sin(A+B) but not able to get there from the Law of Sines just yet.

 

[Quinzio]

You can start by puttin two triangles one "beside" the other, then working with geometry, sines and cosines to get the $$sin(\alpha+\beta)$$.

 

[moko58]

I'm attempting to self [URL='https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/']study mathematics[/URL]. Will be starting college next year and recently got a good score on the SAT Math 2 (~700). I bought "Geometry Revisited" by Coxeter and Greitzer, hoping to gain a bit more knowledge of Geometry. I got stuck on the first question! Until now, all the knowledge I have of geometry was from preparing for my SAT test.

This was the first exercise in the book and so far it has only stated the Law of Sines, so I'm assuming that I won't need anything else to show that a=b cos C + c cos B. Especially seeing at it says that in the question! Haha.

Homework Statement

Show that, for any triangle ABC, even if B or C is an obtuse angle, a=b cos C + c cos B. Use the Law of Sines to deduce the "addition formula":

sin(B+C) = sin B cos C + sin C cos B

Homework Equations

Law of Sines is all that has been stated so far.

The Attempt at a Solution

Non-existent as of yet, apart from stating different sides in terms of trig functions of the angles, but getting nowhere.

I understand this question may be quite trivial in the end seeing as it is the first one in the book, but I would appreciate any points in the right direction.

I am also learning and guessing :-

You just now proved that
a = b Cos C + c Cos B

As per Law Of Sines we can replace a by Sin A, b by Sin B and c by Sin C given that all these belong to the same triangle.

So
Sin A = Sin B Cos C + Sin C Cos B

But Sin A can also be represented as Sin (Pi - A)
But Sin (B + C) is also Sin (Pi - A)

So
Sin (B + C) = Sin B Cos C + Sin C Cos B

 

[andrewkg]

Well this is an old post, but its similar to mine. In the question it says even if b or c is obtuse. And the altitude method will not work. Does anyone know or a secondary method. Or a way to show it works for B or C being obtuse.

 

[Mandelbroth]

Well this is an old post, but its similar to mine. In the question it says even if b or c is obtuse. And the altitude method will not work. Does anyone know or a secondary method. Or a way to show it works for B or C being obtuse.

Yay! Thread ressurection!

Consider that $e^{ix}=\cos{x}+i\sin{x}$. Solve for sine in terms of complex exponentials (hint: what is the parity of the cosine function?) and the sine formula is suddenly rather obvious.