To solve linear trigonometric equations, we can apply basic algebraic techniques similar to those used in solving linear equations. For example, consider the equation:

4 sin(θ) - 3 = 1.

To solve this, we treat the trigonometric function, sin(θ), as a variable. First, we isolate sin(θ) by performing algebraic operations. Adding 3 to both sides gives:

4 sin(θ) = 4.

Next, dividing both sides by 4 results in:

sin(θ) = 1.

Now, we can use the unit circle to find the angle θ. The solution to sin(θ) = 1 is:

θ = \(\frac{\pi}{2}\).

For more complex equations, such as:

-2 cos(θ) + √3 = 0,

we follow similar steps. First, isolate cos(θ) by subtracting √3 from both sides:

-2 cos(θ) = -√3.

Then, divide by -2 to isolate cos(θ):

cos(θ) = \(\frac{\sqrt{3}}{2}\).

Next, we identify the angles on the unit circle where cos(θ) equals \(\frac{\sqrt{3}}{2}\). These angles are:

θ = \(\frac{\pi}{6}\) and θ = \(\frac{11\pi}{6}\).

Since the problem specifies finding solutions within the interval [0, 2π], we do not need to add 2πn to our solutions, as this would apply only if the domain were unrestricted. Thus, our final answers are:

θ = \(\frac{\pi}{6}\) and θ = \(\frac{11\pi}{6}\).

By following these steps—isolating the trigonometric function, identifying solutions on the unit circle, and considering the domain—we can effectively solve any linear trigonometric equation.