how do i solve (sin theta - cos theta = 1)?

Answer 1

sin θ - cos θ = 1

Square both sides:

sin² θ - 2 sin θ cos θ + cos² θ = 1

Use the Pythagorean theorem:

1 - 2 sin θ cos θ = 1

Use the sine addition rule:

1 - sin (2θ) = 1

Subtract 1 from both sides, and multiply by -1:

sin (2θ) = 0

Therefore, 2θ = 2πk or 2θ = π+2πk for some integer k. These solutions can be combined, so 2θ = πk for some integer k. Therefore, θ = πk/2. However, since there was a non-invertible step in our solution (squaring both sides), we must now check each of these solutions. There are four solutions mod 2π: θ=2πk, θ=π/2+2πk, θ=π+2πk, and θ=3π/4+2πk. By inspection, only θ=π/2+2πk and θ=π+2πk actually work. Therefore, the solution is θ∈{π/2+2πk: k∈Z}∪{π+2πk: k∈Z}.

Answer 2

Theta=0 Degrees

Answer 3

Better, is there a closed-form solution or do you have to guess (nothing wrong with that, but bad form if there is an analytical solution) ? My guess is you have to guess, and to look at the behavior of the functions from 0 to 2 pi.

From 0 to pi/2, both functions are positive. At pi/2, sin theta =1, cos theta=0, which is a solution. From pi/2 to pi, cos theta is negative. So we ask: for what angle theta is sin theta + |cos theta| =1? At theta=pi, there is another solution (sin theta=0, cos theta=-1). Keep on going thru to 2pi. Have a nice day.

Answer 4

You could write it inthe form
r*sin(x - A) = 1
where LHS = r*sinx*cosA - r*cosx*sinA
So comparing coefficients with the original expression

r*cosA = 1,
r*sinA = 1

r^2 = 2
tanA = 1
r = sqrt(2)
A = 45 deg

sqrt(2) * sin(x - 45) = 1
sin(x - 45) = 1/sqrt(2)
x - 45 = 45, 135
x = 90, 180

Answer 5

The answer can be discovered by reading through your textbook, understanding the formula and steps then performing the mathematical equation yourself.
The fisherman can give you a fish, or he can teach you how to fish. Learn to do things on your own.

Answer 6

I'm not sure the question makes sense. Have you copied it correctly?

Answer 7

what are you solving for?