document.write( "Question 1081732: If $\"+0+%3C=+x+%3C=+90\"$ °, prove that $\"+1+%3C=+sin%28x%29+%2B+cos%28x%29+%3C=+sqrt%282%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #695785 by htmentor(1343) $\"\"$ $\"About$

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To avoid confusion, let's call the angle t
\n" ); document.write( "sin(t) and cos(t) are defined by a point on the unit circle, (x,y) as sin(t) = y, and cos(t) = x
\n" ); document.write( "Using a trig identity, we can write cos(t) as sqrt(1-sin^2(t))
\n" ); document.write( "So f = sin(t) + cos(t) = sin(t) + sqrt(1-sin^2(t)) = y + sqrt(1-y^2)
\n" ); document.write( "Since the hypotenuse is equal to 1, x and y are less than or equal to 1
\n" ); document.write( "The maximum value is obtained when df/dy = 0 -> 1 - y/sqrt(1-y^2) = 0
\n" ); document.write( "Solving for y, we get y =1/sqrt(2)
\n" ); document.write( "So the maximum value is y + sqrt(1-y^2) = 1/sqrt(2) + 1/sqrt(2) = 2/sqrt(2) = sqrt(2)
\n" ); document.write( "And the minimum value is obtained for y = 0 or y = 1, and the expression takes on the value of 1.
\n" ); document.write( "So we have shown that 1<=sin(t)+cos(t)<=sqrt(2) \n" ); document.write( "

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