Question

Find all solutions of the equation in the interval [0, 2).

sin x-1= cos x
Write your answer(s) in radians in terms of π.
If there is more than one solution, separate them with commas.
x=

Answer 1

Answer: The solutions to the equation sin x - 1 = cos x in the interval [0, 2) are x = -π/2 and x = 3π/4, in radians in terms of π.

Step-by-step explanation:sin x - 1 = cos x

Subtracting cos x from both sides:

sin x - cos x - 1 = 0

Using the identity sin(x - π/4) = sin x cos π/4 - cos x sin π/4, we get:

sin(x - π/4) = -1/√2

Taking the inverse sine of both sides, we get:

x - π/4 = -π/4 - π/2 = -3π/4

or

x - π/4 = π/2 + π/4 = π/2

Adding π/4 to both sides:

x = -3π/4 + π/4 = -π/2

or

x = π/2 + π/4 = 3π/4

Note that the interval [0, 2) contains 0, π/2, and π, but none of these values satisfy the equation. Therefore, the solutions in the given interval are:

x = -π/2 and x = 3π/4.

Hence, the solutions to the equation sin x - 1 = cos x in the interval [0, 2) are x = -π/2 and x = 3π/4, in radians in terms of π.