If - are the solutions of sin x-12 in -0-2- and - are the solutions of cos x-32 in -0-2- then

Solutions of sin x= 12 in [0,2π] are nbsp; x=π+π6 and 2π π6 ⇒ x=7π6 and 11π6 ...(1) Solutions of nbsp;cos x= √(3)2 in [0,2π] are x=π±π6=5π6,7π6 ...(2) ...

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Standard XIII

Mathematics

Principal Solution of Trigonometric Equation

If α,β are th...

Question

If $α, β$ are the solutions of $sin x = - \frac{1}{2}$ in $[0, 2 π]$ and $α, γ$ are the solutions of $cos x = - \frac{\sqrt{3}}{2}$ in $[0, 2 π],$ then

α - β = π

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β - γ = π

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α - γ = π

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α + β = 3 π

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Solution

The correct option is D $α + β = 3 π$
Solutions of $sin x = - \frac{1}{2}$ in $[0, 2 π]$ are
$x = π + \frac{π}{6}$ and $2 π - \frac{π}{6}$
$\Rightarrow x = \frac{7 π}{6}$ and $\frac{11 π}{6} . . . (1)$

Solutions of $cos x = - \frac{\sqrt{3}}{2}$ in $[0, 2 π]$ are
$x = π \pm \frac{π}{6} = \frac{5 π}{6}, \frac{7 π}{6} . . . (2)$

From $(1)$ and $(2),$
$α = \frac{7 π}{6}, β = \frac{11 π}{6}, γ = \frac{5 π}{6}$
$∴ α + β = 3 π$
$α - β = - \frac{2 π}{3}$
$β - γ = π$

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Principal Solution of Trigonometric Equation

Standard XIII Mathematics

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If α,β are the solutions of sinx=−12 in [0,2π] and α,γ are the solutions of cosx=−√32 in [0,2π], then

The correct option is D α+β=3πSolutions of sinx=−12 in [0,2π] are x=π+π6 and 2π−π6 ⇒x=7π6 and 11π6 ...(1) Solutions of cosx=−√32 in [0,2π] are x=π±π6=5π6,7π6 ...(2) From (1) and (2), α=7π6, β=11π6, γ=5π6 ∴α+β=3π α−β=−2π3 β−γ=π

If $α, β$ are the solutions of $sin x = - \frac{1}{2}$ in $[0, 2 π]$ and $α, γ$ are the solutions of $cos x = - \frac{\sqrt{3}}{2}$ in $[0, 2 π],$ then