Question

Sin(п/2-x)-cos(п-x)=1

Answer 1

sin((π/2)-x)=cosx

cos(π-x)=-cosx

sin(п/2-x)-cos(п-x)=cosx-(-cosx)=2cosx

2cosx=1

cosx=1/2

x=±arccos(1/2)+2πn, n∈Z

x=±(π/3)+2πn, n∈Z

Answer 2

Let's solve the trigonometric equation:

sin(π/2 - x) - cos(π - x) = 1

First, we need to use trigonometric identities to simplify the equation. The identities we'll use are:

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

Now, substitute these identities into the equation:

cos(x) - (-cos(x)) = 1 cos(x) + cos(x) = 1 2cos(x) = 1

Next, isolate cos(x) by dividing both sides by 2:

cos(x) = 1/2

Now, to find the values of x, we need to find the angles for which the cosine function equals 1/2. Remember that the cosine function has a period of 2π, which means it repeats every 2π radians. The standard angles where the cosine function equals 1/2 are π/3 and 5π/3:

x = π/3 + 2πn or x = 5π/3 + 2πn

where n is an integer (positive, negative, or zero) that represents the number of full 2π rotations around the unit circle.

So the general solutions for the equation are:

x = π/3 + 2πn or x = 5π/3 + 2πn

where n is an integer. These solutions satisfy the given trigonometric equation.