# Law Of Sines, deducing addition formula

• xavierhart
In summary, the conversation is about an individual's attempt to self-study mathematics and their struggles with a question from a book they purchased. The question involves showing that a certain formula holds true for any triangle and using the Law of Sines to deduce another formula. The individual receives some guidance from others in the discussion, but ultimately comes to a solution by using the properties of complex exponentials.
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.

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.

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

xavierhart said:
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

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.

andrewkg said:
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.

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.

## 1. What is the Law of Sines?

The Law of Sines is a mathematical rule that describes the relationship between the sides and angles of a triangle. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is equal to the ratio of the length of another side to the sine of the angle opposite that side.

## 2. How do you use the Law of Sines to deduce an addition formula?

The Law of Sines can be used to deduce an addition formula by setting up two equations using the law and solving for the unknown angle. Once you have the value of the unknown angle, you can then use it to derive the addition formula.

## 3. What is the purpose of using the Law of Sines to deduce an addition formula?

The purpose of deducing an addition formula using the Law of Sines is to simplify the process of solving trigonometric equations. Addition formulas allow us to combine multiple trigonometric functions into one, making it easier to evaluate and manipulate them.

## 4. Can the Law of Sines be applied to any triangle?

Yes, the Law of Sines can be applied to any triangle, regardless of its shape or size. However, it is important to note that the law only works for triangles, not other shapes.

## 5. Is the Law of Sines the only way to deduce an addition formula?

No, there are multiple ways to deduce an addition formula, including the Law of Cosines and the double-angle formula. However, the Law of Sines is often the most straightforward and efficient method for deducing addition formulas.

Replies
8
Views
5K
Replies
5
Views
2K
Replies
21
Views
3K
Replies
16
Views
2K
Replies
5
Views
2K
Replies
14
Views
691
Replies
7
Views
721
Replies
2
Views
1K
Replies
8
Views
2K
Replies
11
Views
3K