Solve the trigonometric equation over the interval [0, 2π)
[tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]
Show your work, thanks!​

Answers

Answer 1

Answer:

θ = 0° , 60°

Step-by-step explanation:

[tex]\cos^2\theta + \sqrt{3}\sin \theta \cos \theta = 1[/tex]

Substract both the sides from 1.

[tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 1 - 1[/tex]

[tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 0[/tex]

Rearrange the terms in L.H.S.

[tex]=> (\cos^2 \theta - 1 )+ \sqrt{3} \sin \theta \cos \theta = 0[/tex]

Replace the terms in brackets with '-sin²θ' by using the identity (sin²θ + cos²θ = 1).

[tex]=> -\sin^2 \theta + \sqrt{3} \sin \theta \cos \theta = 0[/tex]

Take 'sinθ' common from the whole expression in L.H.S.

[tex]=> \sin \theta(- \sin \theta + \sqrt{3}\cos \theta) = 0[/tex]

Now solve each part separately :-

1) [tex]\sin \theta = 0[/tex]

[tex]=> \theta = \sin^{-1}0 = 0[/tex]

2) [tex]-\sin \theta + \sqrt{3} \cos \theta = 0[/tex]

Divide both the sides by 'cosθ'.

[tex]=> \frac{-\sin \theta + \sqrt{3}\cos \theta }{\cos \theta} = \frac{0}{\cos \theta}[/tex]

[tex]=> \frac{-sin \theta}{\cos \theta} + \frac{\sqrt{3} \cos \theta}{\cos \theta} = 0[/tex]

[tex]=> -\tan \theta + \sqrt{3} = 0[/tex]

Add both the sides with 'tanθ'.

[tex]=> -\tan \theta + \sqrt{3} + \tan \theta = 0 + \tan \theta[/tex]

[tex]=> \tan \theta = \sqrt{3}[/tex]

[tex]=> \theta = \tan^{-1}\sqrt{3} = 60[/tex]

∴ θ = 0° , 60°

Answer 2

Answer:

Solution given:

[tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]

[tex] 1-sin^{2} \theta+\sqrt{3} sin \theta cos \theta = 1[/tex]

[tex] 1-1 +sin^{2}\theta=\sqrt{3} sin \theta cos \theta = 1[/tex]

[tex] Sin\theta(sin\theta -\sqrt{3}cos\theta)[/tex]

Either

[tex] sin\theta=\sqrt{3} cos \theta [/tex]

[tex] \sqrt{sin\theta}{cos \theta}=\sqrt{3} [/tex]

[tex] Tan \theta =\sqrt{3}[/tex]

[tex] Tan \theta =Tan 60,Tan(180+60),Tan (360+60)[/tex]

[tex] \theta =60°,240°,420°[/tex]

in terms of π is

[tex] \theta =⅓π,\frac{4}{3}π,[/tex]

Or

[tex] Sin\theta=1[/tex]

[tex]Sin\theta=Sin0° ,Sin180°[/tex]

In terms of π is

[tex]\theta=0π,π[/tex]

so

[tex]\theta=0π,⅓π,π,\frac{4}{3}π,[/tex]


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Answers

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Step-by-step explanation:

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Answers

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Answers

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Step-by-step explanation:

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Answers

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Answers

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Answers

Step-by-step explanation:

Given :-

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Step-by-step explanation:

Item 3
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Answers

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Answer:

$48

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Cora earns $20 for each lawn

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Answers

Answer:

Step-by-step explanation:

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Answers

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Step-by-step explanation:

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the solution is (2, -6)

Step-by-step explanation:

Substitute the second equation into the first, replacing y in the first:

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Simplifying, we get:

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Answers

Answer:

There won't be enough arrangement

Step-by-step explanation:

Given

[tex]Letters = 2[/tex]

[tex]Digits= 3[/tex]

[tex]Cars = 800000[/tex]

Required

Determine if the possible arrangement will be enough

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Answers

Answer:

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Step-by-step explanation:

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Answers

Answer:

Step-by-step explanation:

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Step-by-step explanation:

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Answer:

Domain = [-5, -2) U (1, 3)

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Step-by-step explanation:

Domain is the x values

Domain = [-5, -2) U (1, 3)

Range is the y values

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Answers

Answer: it’s B, 0.031

Step-by-step explanation: edge 2021

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