Physics, asked by amishasompura2005, 1 year ago

if sinФ +cosФ= √2 cosФ, (Ф≠90°) then value of tan is ?

Answers

Answered by Anonymous
6

Given :

 \bf \sin \theta +  \cos \theta =  \sqrt{2}  \cos\theta

To Find :

   \bf Value  \: of  \: \tan\theta

Solution :

Method 1

Both side divided by Cos Ø

\implies \bf\dfrac{ \sin \theta +  \cos \theta}{ \cos \theta}  =  \dfrac{ \sqrt{2} \cos \theta }{ \cos \theta} \\  \\\bf \implies\dfrac{ \sin \theta}{ \cos \theta} +  \dfrac{ \cancel{ \cos  \theta}}{\cancel{ \cos \theta}}   =  \dfrac{ \sqrt{2}\cancel{ \cos \theta }}{ \cancel{\cos \theta} } \\  \\ \bf \implies \tan \theta  + 1 =  \sqrt{2} \\  \\ \large \implies\boxed{ \bf \tan\theta=  \sqrt{2}  - 1}

Method 2

Squaring both sides

\bf \implies{(\sin\theta + \cos \theta)}^{2} = {(\sqrt{2}\cos\theta)}^{2}\\ \\ \implies \bf {\sin\theta}^{2}+ {\cos\theta}^{2}+2\sin\theta\cos\theta = {2 \cos\theta}^{2}\\ \\ \bf \implies 1 + 2 \sin\theta\cos\theta = {2 \cos\theta}^{2}\\ \\ \bf \implies \sin2\theta= \cos2\theta \\ \\ \bf \implies \sin(2\theta)= \cos(90 - 2\theta )\\ \\ \bf \implies 2 \theta = 90 - 2\theta \\ \\\bf \implies 4 \theta = 90 \\ \\ \bf \implies\theta = 22.5 \\ \\\large \implies\boxed{ \bf \tan\theta=  \sqrt{2}  - 1}

Answered by Anonymous
28

Answer:

Given:

sinθ + cosθ = √2cosθ

to find:

value of tanθ

solution:

squaring both sides-->

(sinθ + cosθ)^2 = (√2cosθ)^2

sinθ^2 + cosθ ^2 +2sinθcosθ= 2cosθ^2

=> 1+ 2sinθcosθ = 2cosθ^2

=> sin2θ = cos2θ

=>2θ = (90-2θ )

=> 4θ = 90

=>θ = 22.5°

=> tan θ = √2-1

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