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Let α = sin⁻¹(√3/2) + sin⁻¹(1/3) and β = cos⁻¹(√3/2) + cos⁻¹(1/3). Which of the following is true?

1. α < β
2. α = β
3. α > β
4. α + β = 2π

Correct answer: α < β

Solution

alpha = pi/3 + arcsin(1/3), beta = pi/6 + arccos(1/3). Using arcsin(1/3)+arccos(1/3)=pi/2, alpha-beta = 2*arcsin(1/3) - pi/3 = -0.37 < 0. So alpha < beta (and alpha+beta = pi, not 2pi).

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