Question

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

Answer 1

SOLUTION

TO DETERMINE

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

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

EVALUATION

Here it is given that

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

$\displaystyle \sf{ \implies \: \sin \theta = \pm \: \frac{ \sqrt{3} }{2} }$

Case : 1

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

$\displaystyle \sf{ \implies \: \sin \theta = \sin {60}^{ \circ} }$

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

Case : 2

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

$\displaystyle \sf{ \implies \: \sin \theta = \sin {60}^{ \circ} }$

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

$\displaystyle \sf{ \implies \: \sin \theta = \sin {120}^{ \circ} }$

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

Case : 3

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

$\displaystyle \sf{ \implies \: \sin \theta = - \sin {60}^{ \circ} }$

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

$\displaystyle \sf{ \implies \: \sin \theta = \sin{240}^{ \circ} }$

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

Case : 4

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

$\displaystyle \sf{ \implies \: \sin \theta = - \sin {60}^{ \circ} }$

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

$\displaystyle \sf{ \implies \: \sin \theta = \sin {300}^{ \circ} }$

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

FINAL ANSWER

θ = 60° , 120° , 240° , 300°

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