Math, asked by BrainlyHelper, 1 year ago

Evaluate each of the following sin² 30° + sin² 45° + sin² 60° + sin² 60° + sin² 90°

Answers

Answered by nikitasingh79
16

SOLUTION :

Given :  

sin² 30° + sin² 45° + sin² 60° + sin² 90°                            

= (½)² + (1/√2√)² + (√3/2)² + (1)²

= ¼ + ½ + ¾ + 1

= ¼ + ¾ + ½ + 1

= (1 + 3)/4 + (1+ 2)/2

[By taking L.C.M]

= 4/4 + 3/2

= 1 + 3/2

= (2 + 3)/2 = 5/2

[ By taking L.C.M]

sin² 30° + sin² 45° + sin² 60° + sin² 90° = 5/2

[sin 60° = √3/2 ,  sin 45° =1/√2  , sin 90° = 1]                                    

Hence, sin² 30° + sin² 45° + sin² 60° + sin² 90° = 5/2

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Answered by abhi569
4

Given Equation : sin² 30° + sin² 45° + sin² 60° + sin² 60° + sin² 90°


         Solution : -

Method 1 ( By using identities )


⇒ sin² 30° + sin² 45° + sin² 60° + sin² 60° + sin² 90°

⇒ sin²30 + sin²45 + sin²( 90 - 30 ) + sin²60 + sin²90

⇒ sin²30 + sin²45 + cos²30 + sin²60 + sin²90

sin²30 + cos²30 + sin²45 + sin²60 + sin²90

\bold{1} + \bigg( \dfrac{1}{\sqrt{2}} \bigg)^{2} + \bigg(\dfrac{\sqrt{3}}{2} \bigg)^{2} + ( 1 )^{2}


⇒ 1 + \dfrac{1}{2} + \dfrac{3}{4} + 1


⇒ 1 + 1 + \dfrac{2+3}{4}


⇒ 2 + \dfrac{5}{4}


\dfrac{8+5}{4}


\dfrac{13}{4}


   

Method 2 ( By using trigonometric table only )


⇒ sin² 30° + sin² 45° + sin² 60° + sin² 60° + sin² 90°

⇒ sin² 30° + sin² 45° + 2sin² 60° + sin² 90°

\bigg( \dfrac{1}{2} \bigg)^{2}+ \bigg(\dfrac{1}{\sqrt{2}} \bigg)^{2} +2 \bigg( \dfrac{\sqrt{3}}{2} \bigg)^{2} + 1


\dfrac{1}{4} +\dfrac{1}{2} + \dfrac{3}{2} + 1


\dfrac{1 + 2 + 6 + 4}{4}


\dfrac{13}{4}



Therefore the value of sin² 30° + sin² 45° + sin² 60° + sin² 60° + sin² 90° is \dfrac{13}{4}

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