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Direction Cosines

If α,β,γ∈[0...

Question

If $α, β, γ \in [0, 2 π]$ , then the sum of all possible values of $α, β, γ$ if $sin α = - \frac{1}{\sqrt{2}}$ , $cos β = - \frac{1}{2}$ , $tan γ = - \sqrt{3}$ , is

A

\frac{22 π}{3}

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B

\frac{21 π}{3}

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C

\frac{20 π}{3}

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D

8 π

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Solution

The correct option is A $\frac{22 π}{3}$
$sin α = \frac{- 1}{\sqrt{2}}$
(-ve) value lies on III, IV quadrant for sin
$∴ α = π + \frac{π}{4}, 2 π - \frac{π}{4}$
$= \frac{5 π}{4}, \frac{7 π}{4}$
$cos β = \frac{- 1}{2}$
(-ve) value lies on II, III quadrant for cosine
$β = π - \frac{π}{3}, π + \frac{π}{3}$
$= \frac{2 π}{3}, \frac{4 π}{3}$
$tan γ = - \sqrt{3}$
(-ve) value lies on II, IV quadrant for tan : $γ = - \sqrt{3}$
$γ = π - \frac{π}{3}, 2 π - \frac{π}{3}$
The sum of all values $(α + β + γ) = \frac{2 π}{3}, \frac{5 π}{3}$
$= \frac{5 π}{4} + \frac{7 π}{4} + \frac{2 π}{3} + \frac{4 π}{3} + \frac{2 π}{3} + \frac{7 π}{3}$
$= 3 π + 2 π + \frac{7 π}{3}$
$= 5 π + \frac{7 π}{3}$
$= \frac{22 π}{3}$

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