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Geometry Level 3

$\large \begin{cases} \cos x + \cos y = \dfrac {\sqrt 3+\sqrt 2}2 \\ \sin x + \sin y = \dfrac {\sqrt 2 + \sqrt 1}2 \end{cases}$

Given the above and that $x$ and $y$ are acute angles, find $x$ and $y$ in degrees and enter your answer as $\frac{\sqrt {2xy}}{\sqrt { 27} }$ .

The answer is 10.

1 solution

Hana Wehbi
Aug 9, 2017

It is so easy to see that $x= 30^\circ$ and $y= 45^\circ$ because $\cos 30^\circ =\frac{ \sqrt{3}}{2}, \sin 30^\circ = \frac{1}{2}$ and $\cos 45^\circ =\frac{ \sqrt{2}}{2}, \sin 45^\circ = \frac{\sqrt{2}}{2}\implies \frac {\sqrt{2xy}}{\sqrt{27}}= 10$

Thank you v. much, Mrs. Hana Nakkache : for your fine solution. in fact you are lucky , but we need an algebraic solution. your methodology is just you solve by guessing or may be by trial and error.I mean how can you solve the problem if the angles are not special angles like 45 and 30, i.e like 40 and 35 also their sum is 75 ....!, but at the end of the day your answer is correct. good luck. thanks.

Aziz Alasha - 3 years, 10 months ago

You have only demonstrated that x=30, y=45 gives the answer of 10, but how do you know that there doesn't exist another pair of (x,y) such that (sqrt(2xy))/sqrt(27) is not equal to 10?

Pi Han Goh - 3 years, 10 months ago

Equal or not equal to $10$ ?

Hana Wehbi - 3 years, 10 months ago

It was clearly stated that x& y are acute angles . So if there is an other pair (x,y) as a solution , it will in the other quadrants, Which are not needed .

Aziz Alasha - 3 years, 10 months ago

Invistigate

The answer is 10.

1 solution

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