Find the number of solutions of the equation sin 2 -cos 2 -4 sin-1-4 cos- lying in the interval -2 - 2 - -

2sin θ cos θ + 1 2sin^2 θ + 4sin θ = 1 + 4cos θ ⇒ 2sin θ(cosθ sin θ) 4(cos θ sin θ) = 0 ⇒ (2sin θ 4) (cos θ sin θ) = 0 But sin θ≠ 2 ∴ tan θ = 1 ⇒θ ...

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Standard XII

Mathematics

General Solution of tan theta = tan alpha

Find the numb...

Question

Find the number of solutions of the equation $s i n 2 θ + c o s 2 θ + 4 s i n θ = 1 + 4 c o s θ$ lying in the interval $[- 2 π, 2 π] .$ ___

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Solution

$2 s i n θ c o s θ + 1 - 2 s i n^{2} θ + 4 s i n θ = 1 + 4 c o s θ$

$\Rightarrow 2 s i n θ (c o s θ - s i n θ) - 4 (c o s θ - s i n θ) = 0$

$\Rightarrow (2 s i n θ - 4) (c o s θ - s i n θ) = 0$

But $s i n θ \neq 2$

$∴ t a n θ = 1 \Rightarrow θ = n π + \frac{π}{4}, n ϵ I$

$∴$ Number of solutions in $[- 2 π, 2 π] are 4, i . e ., θ = - \frac{7 π}{4} - \frac{3 π}{4}, \frac{π}{4}, \frac{5 π}{4}$

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