1 + sin x = cos x?

Question

How do I solve this? Thanks.

Please explain in details. Thanks.

Thank you for the answer but Why is it 0 or 360??

Answer 1

square both side:
1+2sin x+(sin x)^2=(cos x)^2

because (sin x)^2+(cos x)^2=1...

1+2sin x+(sin x)^2 = 1-(sin x)^2
2sin x+2(sin x)^2=0
sin x(1+sin x)=0
sin x=0 or sin x =-1

So possible solutions are 0, 180, 360 and 270 degrees.
But because of the squaring, some solutions might not be valid.

Plug in the values and 180 is not a valid solution,
so I think the answers are 0, 270 and 360 degrees

Answer 2

Actually, those aren't the only solutions. Consider that:
1+sin x = cos x
1 = cos x - sin x
1² = (cos x - sin x)²
1 = cos² x + sin² x - 2 cos x sin x
1 = 1 - 2 sin (2x)
sin (2x) = 0
2x = πk for some integer k
x = πk/2 for some integer k

So all possible solutions are coterminal to one of 0, π/2, π, or 3π/2 radians. However, note that not all of our manipulations were invertible - specifically, the relation sin (2x) = 0 might still be obtained if cos x - sin x = -1 (since (-1)² also equals 1). Therefore we must test these solutions:

1 + sin 0 = cos 0 → 1 + 0 = 1, so 0 is a valid solution
1 + sin (π/2) = cos (π/2) → 1 + 1 = 0, so π/2 is not a valid solution
1 + sin π = cos π → 1 + 0 = -1, so π is not a valid solution
1 + sin (3π/2) = cos (3π/2) → 1 + (-1) = 0, so 3π/2 is a valid solution.

Thus the solution set is: {x| ∃k∈Z : x+2πk = 0 or x+2πk=3π/2}, or in english, all angles coterminal to either 0 or 3π/2.

Answer 3

What do you mean "solve" it? For the value of x?

1 = cos x - sin x. This occurs at 0 and 3/2 pi +/- k2pi, where k is any integer, including 0.

Why 3/2 pi? Well, this corresponds to 1/2pi (90 deg) + pi. At 90 deg, cos = 0, sin = 1. However, when this is shifted pi radians (or 180 deg), then you get cos = 0, sin = -1. (3/2pi rad = 270 deg)

Answer 4

0 degrees or 360 degrees

i don't think there is a way to solve it mathematically, but if u graph it in a calculator with the equation 1+sin x -cos x=0
y=0 when x=0

1+sin(0)=cos(0)=1

Answer 5

x = 0 degrees.

sin(0) = 0, cos(0)=1
1+0=1