SOLUTION: what is the minimum & maximum value of sin(x)+cos(x)
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Question 1046265: what is the minimum & maximum value of sin(x)+cos(x)
Answer by ikleyn(52754) (Show Source): You can put this solution on YOUR website!
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what is the minimum & maximum value of sin(x)+cos(x)
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The maximum is .
The minimum is .
There are many different ways to prove it.
Below is one of the ways.
Let y = sin(x) + cos(x).
Then
y^2 = (sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 1 + 2sin(x)cos(x) = 1 + sin(2x). (1)
So, y, and hence, sin(x) + cos(x) is maximal when sin(2x) = 1.
It happens when 2x = , or x = .
At this value of x, y^2 = 2 and hence y = is the maximum of sin(x) + cos(x).
From (1), <= 2. Hence, |y| <= , or <= y <= .
Finally, sin(x) + cos(x) = at x = .
It proves that the maximum is and the minimum is .
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