Definitely not Descartes' rule of signs
For a triangle A B C , we have sin ( A ) = 0 . 6 0 0 and sin ( B ) = 0 . 9 6 0 .
Which of these answer choices is a possible value of sin ( C ) ?
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1 solution
Yay! I'm sure you enjoyed this! I took a long time picking out the best few numbers to use in MCQ.
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Yup! A really nice one!
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My solution: Draw an obtuse isosceles triangle with angles sin − 1 ( 0 . 6 ) , sin − 1 ( 0 . 6 ) , π − 2 sin − 1 ( 0 . 6 ) , apply sin ( π − A ) = sin ( A ) and sin ( 2 A ) = 2 sin ( A ) cos ( A ) .
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@Pi Han Goh – But for a person who has read the question for the first time, how can he determine that the triangle is going to be isoceles and obtuse? He can make out that the triangle is going to be either acute or obtuse angled. But he has to consider both your options and the different cases.
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@Vishnu C – My solution basically says that you need to know the answer beforehand. Or you can just plug in all the 4 values given and show that one of it actually works!
Another way to see it is that the sine of one angle is 0.600 while the sine of twice that same angle is 0.960, the rest should be obvious.
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@Pi Han Goh – If you know the answer beforehand, your solution must be like this:
I knew the answer beforehand! Haha! Didn't have to calculate it! ; D
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@Vishnu C – But you still have to prove that your solution works! ;)
I knew that this question is gonna be tricky and I was right. Most MCQ questions on brilliant are tricky ones so that you can't go back to recheck your thinking.
Anyway, here's how I did it:
sin ( A ) = 0 . 6 0 0 ⇒ cos ( A ) = ± 0 . 8 0 0 .
sin ( B ) = 0 . 9 6 0 ⇒ cos ( B ) = ± 0 . 2 8 .
sin ( A + B ) = sin ( A ) cos ( B ) + sin ( B ) cos ( A ) = sin ( C ) since A + B + C = π .
sin ( C ) cannot be negative. So the cosines of A and B cannot both be negative. And that would also mean that both angles A and B are obtuse, which cannot happen.
If both are positive, you get sin ( C ) = 0 . 9 3 6 , which is not an option (although 0.935 is).
Then the only other case is when cos(B) is negative. In this case, the answer is 0.6, which is an option. And in this case, angle B is obtuse.
This method may seem a bit silly and I might have looked over a result that would have made the method shorter. But this is how I did it. Tricky question!