SATHEE: Trigonometrical Equations 1 Question 6

Trigonometrical Equations 1 Question 6

6. The sum of all values of $θ \in 0, \frac{π}{2}$ satisfying $\sin^{2} 2 θ + \cos^{4} 2 θ = \frac{3}{4}$ is

(a) $\frac{3 π}{8}$

(b) $\frac{5 π}{4}$

(d) $π$

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Answer:

Correct Answer: 6. (c)

Solution:

Given, $\sin^{2} 2 θ + \cos^{4} 2 θ = \frac{3}{4}$

$\Rightarrow (1 - \cos^{2} 2 θ) + \cos^{4} 2 θ = \frac{3}{4} (∵ \sin^{2} x = 1 - \cos^{2} x)$

$\Rightarrow 4 \cos^{4} 2 θ - 4 \cos^{2} 2 θ + 1 = 0$

$\Rightarrow {(2 \cos^{2} 2 θ - 1)}^{2} = 0$

$\Rightarrow 2 \cos^{2} 2 θ - 1 = 0 \Rightarrow \cos^{2} 2 θ = \frac{1}{2}$

$\Rightarrow \cos 2 θ = \pm \frac{1}{\sqrt{2}}$

If $θ \in 0, \frac{π}{2}$ , then $2 θ \in (0, π)$

$∴ \cos 2 θ = \pm \frac{1}{\sqrt{2}}$

$\Rightarrow 2 θ = \frac{π}{4}, \frac{3 π}{4}$ ,

$∵ \cos \frac{3 π}{4} = \cos π - \frac{π}{4} = - \cos \frac{π}{4} = - \frac{1}{\sqrt{2}}$

$\Rightarrow θ = \frac{π}{8}, \frac{3 π}{8}$

Sum of values of $θ = \frac{π}{8} + \frac{3 π}{8} = \frac{π}{2}$

Trigonometrical Equations 1 Question 6

6. The sum of all values of $θ \in 0, \frac{π}{2}$ satisfying $\sin^{2} 2 θ + \cos^{4} 2 θ = \frac{3}{4}$ is

Answer:

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Trigonometrical Equations 1 Question 6

6. The sum of all values of θ∈0,π2 satisfying sin2⁡2θ+cos4⁡2θ=34 is

Answer:

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6. The sum of all values of $θ \in 0, \frac{π}{2}$ satisfying $\sin^{2} 2 θ + \cos^{4} 2 θ = \frac{3}{4}$ is