Question

solve 2 sin square theta - 4 equal to 5 cos theta

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Answer 1

Step-by-step explanation:

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Answer 2

Given:

2$\sin^2 \theta$ - 4 = 5$\cos \theta$

To solve 2$\sin^2 \theta$ - 4 = 5$\cos \theta$.

Solution:

∴ 2$\sin^2 \theta$ - 4 = 5$\cos \theta$

Using the trigonometric identity:

$\sin^2 \theta$ + $\cos^2 \theta$ = 1

⇒ $\sin^2 \theta$ = 1 - $\cos^2 \theta$

2(1 - $\cos^2 \theta$ ) - 4 = 5$\cos \theta$

⇒ 2 - 2$\cos^2 \theta$ - 4 = 5$\cos \theta$

⇒ - 2$\cos^2 \theta$ - 2 = 5$\cos \theta$

⇒ 2$\cos^2 \theta$ + 5$\cos \theta$ + 2 = 0

By factorisation method,

2$\cos^2 \theta$ + 4$\cos \theta$ + $\cos \theta$ + 2 = 0

⇒ 2$\cos \theta$($\cos \theta$ + 2) + 1($\cos \theta$ + 2) = 0

⇒ (2$\cos \theta$ + 1)($\cos \theta$ + 2) = 0

⇒ 2$\cos \theta$ + 1 = 0 or, $\cos \theta$ + 2 = 0

⇒ 2$\cos \theta$ + 1 = 0 or, $\cos \theta$ + 2 = 0

⇒ 2$\cos \theta$ = - 1 or, $\cos \theta$ = - 2

⇒ $\cos \theta$ = $\dfrac{-1}{2}$ [ ∵ $\cos \theta$ = - 2 is not possible]

∴ $\cos \theta$ = $\dfrac{-1}{2}$

⇒ $\cos \theta$ = $\cos 120$°

⇒ θ = 120°

Thus, θ = 120°