Question #5448

Hello Dear Expert what isthe greatest value of "sin theta +cos theta"

Expert's answer

the first step – take the first derivative of the function (sin theta + cos theta):

d f / d theta = -cos theta + sin theta =>

=> in max/min point d f / d theta = 0 =>

=> sin theta = cos theta => divide both parts by cos theta => tg theta = 1 =>

=> theta = pi/4 + n*pi, where n – integer =>

=> so we have theta = 45^{0}, sin 45^{0} = (sqrt(2)/2), cos 45^{0} =(sqrt(2)/2), sin 45^{0 }+ cos 45^{0 }= sqrt(2) ≈ 1,414

Check if theta = 45^{0} is not the minimum point, take the other value of theta = 0^{0}, sin 0 + cos 0 = 0 + 1 = 1 < sqrt(2), thus** greatest value of (sin theta + cos theta = 1.414)**

d f / d theta = -cos theta + sin theta =>

=> in max/min point d f / d theta = 0 =>

=> sin theta = cos theta => divide both parts by cos theta => tg theta = 1 =>

=> theta = pi/4 + n*pi, where n – integer =>

=> so we have theta = 45

Check if theta = 45

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