Question

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find all the angles between 0 and 360 which satisfy the equation sin^2theta=3/4?​

Answer 1

Answer:

SOLUTION

TO DETERMINE

All the angles θ where 0° < θ < 360° and

\displaystyle \sf{ { \sin }^{2} \theta = \frac{3}{4} }sin

2

θ=

4

3

EVALUATION

Here it is given that

\displaystyle \sf{ { \sin }^{2} \theta = \frac{3}{4} }sin

2

θ=

4

3

\displaystyle \sf{ \implies \: \sin \theta = \pm \: \frac{ \sqrt{3} }{2} }⟹sinθ=±

2

3

Case : 1

\displaystyle \sf{ \sin \theta = \: \frac{ \sqrt{3} }{2} }sinθ=

2

3

\displaystyle \sf{ \implies \: \sin \theta = \sin {60}^{ \circ} }⟹sinθ=sin60

∘

\displaystyle \sf{ \implies \: \theta = {60}^{ \circ} }⟹θ=60

∘

Case : 2

\displaystyle \sf{ \sin \theta = \: \frac{ \sqrt{3} }{2} }sinθ=

2

3

\displaystyle \sf{ \implies \: \sin \theta = \sin {60}^{ \circ} }⟹sinθ=sin60

∘

\displaystyle \sf{ \implies \: \sin \theta = \sin( {180}^{ \circ} - {60}^{ \circ}) }⟹sinθ=sin(180

∘

−60

∘

)

\displaystyle \sf{ \implies \: \sin \theta = \sin {120}^{ \circ} }⟹sinθ=sin120

∘

\displaystyle \sf{ \implies \: \theta = {120}^{ \circ} }⟹θ=120

∘

Case : 3

\displaystyle \sf{ \sin \theta = \: - \frac{ \sqrt{3} }{2} }sinθ=−

2

3

\displaystyle \sf{ \implies \: \sin \theta = - \sin {60}^{ \circ} }⟹sinθ=−sin60

∘

\displaystyle \sf{ \implies \: \sin \theta = \sin( {180}^{ \circ} + {60}^{ \circ}) }⟹sinθ=sin(180

∘

+60

∘

)

\displaystyle \sf{ \implies \: \sin \theta = \sin{240}^{ \circ} }⟹sinθ=sin240

∘

\displaystyle \sf{ \implies \: \theta = {240}^{ \circ} }⟹θ=240

∘

Case : 4

\displaystyle \sf{ \sin \theta = - \: \frac{ \sqrt{3} }{2} }sinθ=−

2

3

\displaystyle \sf{ \implies \: \sin \theta = - \sin {60}^{ \circ} }⟹sinθ=−sin60

∘

\displaystyle \sf{ \implies \: \sin \theta = \sin( {360}^{ \circ} - {60}^{ \circ}) }⟹sinθ=sin(360

∘

−60

∘

)

\displaystyle \sf{ \implies \: \sin \theta = \sin {300}^{ \circ} }⟹sinθ=sin300

∘

\displaystyle \sf{ \implies \: \theta = {300}^{ \circ} }⟹θ=300

∘

FINAL ANSWER

θ = 60° , 120° , 240° , 300°

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