Can you take the square roots out?

Geometry Level 4

$\large \sqrt{\sin 3\theta + \sin 2\theta} -\sqrt{\sin 3\theta - \sin 2\theta} = \sin 3\theta$

Given that $0^\circ < \theta < 90^\circ$ , and $\theta$ is a whole number, what is the value of $\sin 3\theta$ ?

The answer is 1.

1 solution

Sahil Silare
Dec 16, 2016

This problem can be solved by hit and trail. As $0^\circ < \theta < 90^\circ$ , I took a value $30^\circ$ then $sin3\theta$ becomes $sin90^\circ$ which is equal to 1.

If you square both sides and write both sides in terms of sin(theta), you can get a polynomial equation (squaring again) which has solution 1/2, leading to the given solution. I have more work to do to see if there are other non-extraneous possibilities. If the difference of the two sides is monotonic, that would show uniqueness, something I have not had the time to try.

Will Heierman - 4 years, 6 months ago

Yeah we can try that too.

Sahil Silare - 4 years, 6 months ago

I worked out the equation. If there are other values of u = cos(theta), they would be roots of the polynomial 128u^9 + 64u^8 - 352u^7 - 48u^6 + 408u^5 - 84u^4 - 258u^3 + 87u^2 + 76u - 16. Someone with a factoring algorithm might be able to complete the analysis.

Will Heierman - 4 years, 5 months ago

But why are you making more complications to the problem ? We always find an easier way to solve problems in mathematics.

Sahil Silare - 4 years, 5 months ago

Sahil, the statement implies the equation has a unique solution. Finding one solution does not prove there are not others. My remarks dealt with this uncertainty.

Will Heierman - 4 years, 4 months ago

Listen what I have done satisfies the given conditions right? So its not uncertain.

Sahil Silare - 4 years, 4 months ago

See by logic we know sin 3theta is integer bcuz the question states that, now theta ≠0 So theta might be 1 or - 1, now try both of these as u have 3 answers and u get ur answer

Topper Forever - 4 years, 3 months ago

Also theta is bw 0 and 90 so answer is 1, Understand everybody, I am the next imo gold medalist

Topper Forever - 4 years, 3 months ago

Oh very cool NEXT IMO GOLD MEDALIST ! LOL

Sahil Silare - 4 years, 3 months ago

Can you take the square roots out?

The answer is 1.

1 solution

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