SOLUTION: what is the minimum & maximum value of sin(x)+cos(x)

Algebra ->  Trigonometry-basics -> SOLUTION: what is the minimum & maximum value of sin(x)+cos(x)      Log On


   

Question 1046265: what is the minimum & maximum value of sin(x)+cos(x)
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
what is the minimum & maximum value of sin(x)+cos(x)
~~~~~~~~~~~~~~~~~~~~~~~~~~~

The maximum is sqrt%282%29.

The minimum is -sqrt%282%29.

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 = pi%2F2,  or  x = pi%2F4.

At this value of  x,  y^2 = 2  and hence  y = sqrt%282%29  is the maximum of  sin(x) + cos(x).

From (1),  y%5E2 <= 2.  Hence,  |y| <= sqrt%282%29, or -sqrt%282%29 <= y <= sqrt%282%29.

Finally,  sin(x) + cos(x) = -sqrt%282%29  at  x = 5pi%2F4.

It proves that the maximum is  sqrt%282%29  and the minimum is  -sqrt%282%29.



Plot y = sin(x) + cos(x)