inspiration:- Mistakes give rise to problems!

Geometry Level 4

$\large\sin(x+y)=\sin(x)+\sin(y)$ We know that the above splitting is not allowed. This is a false property, it will be a mistake if you do it! But for how many ordered pairs $(x,y)$ , where $x$ and $y$ are integral multiples of $\pi$ and $-\pi\leq x,y\leq 3\pi$ , does the above equality seem to hold true?

Inspired by Aditya Raut 's set Mistakes Give Rise To Problems! .

The answer is 25.

2 solutions

Chew-Seong Cheong
Jun 24, 2015

The equation is true when $x$ and $y$ are integer multiples of $\pi$ . There are $5$ integer multiples of $\pi$ in $[-\pi, 3\pi ]$ . Therefore, there are $5$ values of acceptable $x$ and $5$ values of acceptable $y$ , a total of $5 \times 5 = \boxed{25}$ ordered pairs of $(x,y)$ .

Same method used.

Niranjan Khanderia - 3 years, 5 months ago

Aran Pasupathy
Jun 24, 2015

With the given conditions, there are 5 possible angles that x and y can assume, which are -π, 0, π, 2π and 3π.

Now there are ((5!)/(5-2)!) ways of permuting two out of the five possible values, which is equal to 20. But we also need to account for the pairs for which x and y have the same value. There are 5 such pairs.

There are thus 25 ordered pairs (x,y) for which the above equation appears to hold true.

inspiration:- Mistakes give rise to problems!

Inspired by Aditya Raut 's set Mistakes Give Rise To Problems! .

The answer is 25.

2 solutions

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