Inverse Dilemma

Geometry Level 3

{ cos ⁡ − 1 x + cos ⁡ − 1 y + cos ⁡ − 1 z = 0 sin ⁡ − 1 x + cos ⁡ − 1 y + tan ⁡ − 1 z = ? \large \begin{cases} \cos^{-1}x + \cos^{-1}y + \cos^{-1}z = 0 \\ \\ \sin^{-1}x + \cos^{-1} y + \tan^{-1} z = \ ? \end{cases}

Details :

  • − 1 ≤ x , y , z ≤ 1 -1 \leq x , y , z \leq 1
Insufficient Information Not Defined 3 3 3 π 4 \dfrac{3\pi}{4} π 2 \dfrac{\pi}{2} π \pi π 4 \dfrac{\pi}{4} 0 0

Akhil Bansal by Akhil Bansal , 22, India

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1 solution

Akhil Bansal
Oct 8, 2015

Observe that minimum value of cos ⁡ − 1 ( x ) \color{#3D99F6}{\cos^{-1}(x)} function is zero.
Therfore, cos ⁡ − 1 x + cos ⁡ − 1 y + cos ⁡ − 1 z = 0 \cos^{-1} x + \cos^{-1} y + \cos^{-1} z = 0 only when cos ⁡ − 1 x = cos ⁡ − 1 y = cos ⁡ − 1 z = 0 \cos^{-1}x = \cos^{-1} y = \cos^{-1} z = 0
Hence, x = y = z = 1 x \ = \ y \ = \ z = 1

⇒ sin ⁡ − 1 1 + cos ⁡ − 1 1 + tan ⁡ − 1 1 = π 2 + 0 + π 4 = 3 π 4 \Rightarrow \sin^{-1} 1 + \cos^{-1} 1 + \tan^{-1} 1 = \dfrac{\pi}{2} + 0 + \dfrac{\pi}{4} = \dfrac{3\pi}{4}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Perhaps add that x , y , z x, y, z are values from -1 to 1?

Calvin Lin Staff - 5 years, 8 months ago

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