Write the number of values of - in - 2 - that satisfy the equation sin 2-cos-1-4-

We have, sin^2θ cos θ=1/4 ⇒ 1 cos^2θ cos θ=1/4 ⇒ cos^2θ+cos θ 3/4=0 ⇒ 4cos^2θ+4 cos θ 3=0 ⇒ 4cos^2θ +6 cos θ 2 cos θ 3=0 ⇒ 2 cos θ[2 cos θ+3] [2 cos θ+3]=0 ⇒ ( ...

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Standard XII

Mathematics

Equations with Solutions at Boundary Values

Write the num...

Question

$Write the number of values of θ i n [θ, 2 π] t h a t s a t i s f y t h e e q u a t i o n s i n^{2} θ - c o s θ = \frac{1}{4} .$

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Solution

$W e h a v e, s i n^{2} θ - c o s θ = \frac{1}{4} \Rightarrow 1 - c o s^{2} θ - c o s θ = \frac{1}{4} \Rightarrow c o s^{2} θ + c o s θ - \frac{3}{4} = 0 \Rightarrow 4 c o s^{2} θ + 4 c o s θ - 3 = 0 \Rightarrow 4 c o s^{2} θ + 6 c o s θ - 2 c o s θ - 3 = 0 \Rightarrow 2 c o s θ [2 c o s θ + 3] - [2 c o s θ + 3] = 0 \Rightarrow (2 c o s θ - 1) (2 c o s θ + 3) = 0 ∴ c o s θ = \frac{1}{2} o r c o s θ = \frac{- 3}{2} ∴ O n l y f o r 2 v a l u e s i n [0, 2 π], c o s θ = \frac{1}{2}, o r t h e g i v e n e q u a t i o n i s s a t i s f i e d .$

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Write the number of values of θ in [θ,2π]that satisfy the equation sin2 θ−cos θ=14.

$Write the number of values of θ i n [θ, 2 π] t h a t s a t i s f y t h e e q u a t i o n s i n^{2} θ - c o s θ = \frac{1}{4} .$