Math, asked by mailmemurali98, 1 month ago

pls answer me friends​

Attachments:

Answers

Answered by mathdude500
5

Given Question :-

Evaluate :-

\rm :\longmapsto\:1 + cos10 + cos20 + cos30

 \green{\large\underline{\sf{Solution-}}}

Given Trigonometric expression is

\rm :\longmapsto\:1 + cos10 + cos20 + cos30

can be re-arranged as

\rm \:  =  \: (1 + cos30) + (cos20 + cos10)

We know,

\boxed{ \tt{ \: 1 + cos2x =  {2cos}^{2}x \: }} \\  \\ and \\  \\ \boxed{ \tt{ \: cosx + cosy = 2cos\bigg[\dfrac{x + y}{2} \bigg]cos\bigg[\dfrac{x - y}{2} \bigg]}}  \\

So, using this, we get

\rm \:  =  \: 2 {cos}^{2}\dfrac{30}{2}  + 2cos\bigg[\dfrac{20 + 10}{2} \bigg]cos\bigg[\dfrac{20 - 10}{2} \bigg]

\rm \:  =  \: 2 {cos}^{2}15  + 2cos\bigg[\dfrac{30}{2} \bigg]cos\bigg[\dfrac{10}{2} \bigg]

\rm \:  =  \: 2 {cos}^{2}15  + 2cos15 \: cos5

\rm \:  =  \: 2 {cos}^{2}15 + 2cos15 \: cos5

\rm \:  =  \: 2 \: cos15 \: [cos15 + cos5]

\rm \:  =  \: 2 \: cos15 \: \bigg[2 \: cos\dfrac{15 + 5}{2} \: cos\dfrac{15 - 5}{2}  \bigg]

\rm \:  =  \: 4 \: cos15 \: \bigg[ \: cos\dfrac{20}{2} \: cos\dfrac{10}{2}  \bigg]

\rm \:  =  \: 4 \: cos15 \: cos10 \: cos5

\rm \:  =  \: 4 \: cos5 \: cos10 \: cos15

Hence,

\boxed{ \tt{1 + cos10 + cos20 + cos30 =4cos5 \: cos10 \: cos15}}

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information :-

\boxed{ \tt{ \: sinx + siny = 2sin\bigg[\dfrac{x + y}{2} \bigg]cos\bigg[\dfrac{x - y}{2} \bigg]}}

\boxed{ \tt{ \: sinx  -  siny = 2cos\bigg[\dfrac{x + y}{2} \bigg]sin\bigg[\dfrac{x - y}{2} \bigg]}}

\boxed{ \tt{ \: cosx - cosy = -  \:  2sin\bigg[\dfrac{x + y}{2} \bigg]sin\bigg[\dfrac{x - y}{2} \bigg]}}

Similar questions