Given both and are acute angles and sin = 1 2 , cos = 1 3 , then the value of belongs to [2004]

Question

Given both $θ$ and $ϕ$ are acute angles and $\sin θ = \frac{1}{2}, \cos ϕ = \frac{1}{3},$ then the value of $θ + ϕ$ belongs to [2004]

Smart Achievers · Accepted Answer

(2)

$Given: \sin θ = \frac{1}{2} and θ is acute angle$

$∴$ $θ = \frac{π}{6}$

$Also given, \cos ϕ = \frac{1}{3} and ϕ is acute angle.$

$∴$ $0 < \frac{1}{3} < \frac{1}{2}$

$\Rightarrow \cos \frac{π}{2} < \cos ϕ < \cos \frac{π}{3} or \frac{π}{3} < ϕ < \frac{π}{2}$

$∴$ $\frac{π}{3} + \frac{π}{6} < θ + ϕ < \frac{π}{2} + \frac{π}{6} or \frac{π}{2} < θ + ϕ < \frac{2 π}{3}$

$\Rightarrow θ + ϕ \in (\frac{π}{2}, \frac{2 π}{3})$

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