Well!This is a nice problem

If $\sin(A+B)=\sin A+\sin B$ Then how many distinct values of $A,B$ are there?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Jon Haussmann @Brian Charlesworth @Brandon Monsen @Otto Bretscher @Pi Han Goh Any Help

Sumukh Bansal - 3 years, 6 months ago

Infinitely many.

Both $A/\pi$ and $B / \pi$ can be any integer, and the equation will be fulfilled.

Pi Han Goh - 3 years, 6 months ago

^^ This

I'm not sure if this is a complete solution set, but I found that

$A = 2\pi k$ for some integer $k$ will be a solution for all $B$

$A+B = 2\pi k$ for some integer $k$ will be a solution

Brandon Monsen - 3 years, 6 months ago

@Brandon Monsen – I could go into more detail by expanding sin(A+B) = sinA cosB + cosA sinB, but that's not needed. The question only asked for "How many solutions are there?"

Pi Han Goh - 3 years, 6 months ago

@Pi Han Goh – Oh when I said "not sure if this is a complete solution set" I was referring to myself. Yours is perfect for what the problem asks, just figured I'd give more since I'd done it anyways and only took a few seconds to type up

Brandon Monsen - 3 years, 6 months ago

@Pi Han Goh – What if there is a condition that $A,B\le360^\circ$

Sumukh Bansal - 3 years, 6 months ago

@Sumukh Bansal – There's still infinitely many. The condition that $A+B=2\pi$ means that $A$ can take any real number between $0$ and $2\pi$ and there will be a value of $B$ which satisfies the equations

Brandon Monsen - 3 years, 6 months ago

@Brandon Monsen – Yes, after doing the expansion and manipulation I find that either

(i) $\cos(A) = \cos(B) \Longrightarrow A + B = 2\pi k$ or

(ii) either $\cos(A) = 1 \Longrightarrow (A,B) = (2\pi k, B)$ or $\cos(B) = 1 \Longrightarrow (A,B) = (A,2\pi k)$ ,

provides the complete set of solutions.

Brian Charlesworth - 3 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$