Question

Period of sin2x +cos2x ?

Answer 1

The period is `pi`

Explanation:

Let

`y = sin(2x) + cos(2x)`

Then

`y^2 = sin^2(2x) + cos^2(2x) + 2sin(2x)cos(2x)`

`y^2 = 2sin(2x)cos(2x) + 1`

`y^2 = sin(4x) + 1`

`y = +-sqrt(sin(4x) + 1)`

The square root or the horizontal transformation doesn't affect the period, however what we've done is pretty much split up the original function into two functions, the negative and the positive. Therefore, our period will be the double of the period of `+sqrt(sin(4x) + 1)`. Thus period will be `(2pi)/4 *2 = pi`.

We now do a graphical verification.

We can see that the distance between two consecutive maximums (or minimums) is `(9pi)/8 - pi/8 = pi`, as required.

Hopefully this helps!

Answer 2

`pi`

Explanation:

f(x) = sin 2x + cos 2x
Period of sin 2x --> `(2pi)/2 = pi`
Period of cos 2x --> `(2pi)/2 = pi`
Period of f(x) --> `pi` (least common multiple of `pi and pi`)