Question

How do you find the x intercepts of `y=sinpix+cospix`?

Answer 1

`x = k - 1/4" "` for any integer `k`

Explanation:

Note that:

`y = sin pi x + cos pi x`

`color(white)(y) = sqrt(2)(sqrt(2)/2 sin pi x + sqrt(2)/2 cos pi x)`

`color(white)(y) = sqrt(2)(sin (pi/4) sin pi x + cos (pi/4) cos pi x)`

`color(white)(y) = sqrt(2) sin (pi/4 + pi x)`

Also note that:

`sin theta = 0" "` if and only if `theta = k pi` for some integer `k`

So we require:

`pi/4 + pi x = k pi`

Dividing both sides by `pi`, this becomes:

`1/4 + x = k`

Then subtracting `1/4` from both sides we find:

`x = k - 1/4" "` for any integer `k`