The number of values of in the interval ( - 2 , 2 ) such that n 5 for n = 0 , 1 , 2 and tan = cot 5 as well as sin 2 = cos 4 is [2010]

Question

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]

Smart Achievers · Accepted Answer

(3)

$\tan θ = \cot 5 θ, θ \neq \frac{n π}{5}$

$\Rightarrow \cos θ \cos 5 θ - \sin 5 θ \sin θ = 0 \Rightarrow \cos 6 θ = 0$

$\Rightarrow 6 θ = \frac{- 5 π}{2}, \frac{- 3 π}{2}, \frac{- π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}$

$\Rightarrow θ = \frac{- 5 π}{12}, \frac{- π}{4}, \frac{- π}{12}, \frac{π}{12}, \frac{π}{4}, \frac{5 π}{12}$

$Again \sin 2 θ = \cos 4 θ = 1 - 2 \sin^{2} 2 θ$

$\Rightarrow 2 \sin^{2} 2 θ + \sin 2 θ - 1 = 0 \Rightarrow \sin 2 θ = - 1, \frac{1}{2}$

$\Rightarrow 2 θ = \frac{- π}{2}, \frac{π}{6}, \frac{5 π}{6} \Rightarrow θ = \frac{- π}{4}, \frac{π}{12}, \frac{5 π}{12}$

$So, common solutions are θ = \frac{- π}{4}, \frac{π}{12}, \frac{5 π}{12}$

$∴$ $Number of solutions = 3$

Solution

Q.
The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]

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Solution

Q. The number of values of θ in the interval (-π2,π2) such that θ≠nπ5 for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is [2010]

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Q.
The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $\tan θ = \cot 5 θ$ as well as $\sin 2 θ = \cos 4 θ$ is [2010]