Question

Sin(п- x)-cos(п÷2+x)=√3

Answer 1

Sin(x)+sin(x)=√3
2sin(x)=√3
sin(x)=√3/2
x=(-1)^k*pi/3+pi*n, n пренадлежит Z

Answer 2

To solve the trigonometric equation sin(π - x) - cos(π/2 + x) = √3, we will use trigonometric identities and algebraic manipulation. First, we need to express both sin(π - x) and cos(π/2 + x) in terms of trigonometric functions.

1. sin(π - x) = sin(π)cos(x) - cos(π)sin(x) = 0 - (-1)sin(x) = sin(x)

2. cos(π/2 + x) = cos(π/2)cos(x) - sin(π/2)sin(x) = 0 - (1)sin(x) = -sin(x)

Now, the equation becomes:

sin(x) - (-sin(x)) = √3 sin(x) + sin(x) = √3 2sin(x) = √3

Now, we need to isolate sin(x), so we'll divide both sides by 2:

sin(x) = √3 / 2

To find the solutions for x, we need to determine the angles whose sine is √3/2. We know that sin(π/3) = √3/2.

So, we have:

x = π/3 + 2nπ or x = π - π/3 + 2nπ

where n is an integer.

The first set of solutions (π/3 + 2nπ) represents the angles in the first and second quadrants, and the second set of solutions (π - π/3 + 2nπ) represents the angles in the third and fourth quadrants.

To summarize, the solutions for x are:

1. x = π/3 + 2nπ (where n is an integer)
2. x = π - π/3 + 2nπ (where n is an integer)