A geometry problem by Aly Ahmed

Geometry Level 3

5 sin ⁡ x − cos ⁡ x + 5 cos ⁡ x − sin ⁡ x = 2 ( sin ⁡ x + cos ⁡ x ) 5^{\sin x - \cos x} + 5^{\cos x - \sin x} = \sqrt2 (\sin x + \cos x)

Find x x (in degrees) in the range 0 ∘ ≤ x ≤ 9 0 ∘ 0^\circ \leq x \leq 90^\circ satisfying the equation above.


The answer is 45.

Aly Ahmed by Aly Ahmed , 34, Egypt

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

L. H. S. is ≥ 2 5 sin ⁡ x − cos ⁡ x × 5 cos ⁡ x − sin ⁡ x \geq 2 \sqrt {5^{\sin x-\cos x}\times 5^{\cos x-\sin x}} or ≥ 2 \geq 2 . R. H. S. is 2 sin ⁡ ( x + π 4 ) ≤ 2 2\sin \left(x+\dfrac{π}{4}\right) \leq 2 . So the only solution is the value of x x for which each side of the equation is 2 2 . This happens when x = 45 ° x=\boxed {45\degree} .

0 Helpful 0 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...