Help, please to solve this problem

Question

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

2 sin 0(theta)-1=0

Write your answer in radians in terms of pi.

Answer 1

2sin(θ)-1=0

2sin(θ)=1

sin(θ)=1/2

θ= sin^-1(1/2) , i.e. θ is any angle that has a sine of 1/2.

There are infinitely many such angles, but in the interval from [0, 2π) there are exactly two, one in the first quadrant and another in the 2nd quadrant (sine is positive in those quadrants).

In the 30-60-90 right triangle, the sine of the 30 degree angle is 1/2, because the hypotenuse is always twice the short leg. Think of the 30 degree angle as the reference angle (in radians π/6).

In the first quadrant the angle is π/6. In the 2nd quadrant the angle is 180-30 or 6π/6 - π/6 = 5π/6.

Here is a Desmos graph for verification:

desmos.com/calculator/zvwshd8jjj