Solve the Trigonometric equation for all solutions between [0,360)degrees.

2Sinx + 1 = 2.6635.

Solve for x.

does x equal 56.28degrees and 123.72degrees?

Ah, finally a version which can be solved:

2sinx = 1.6635
sinx = 0.8317
x = 56.28° and 180-56.28°

you are correct.

To solve the trigonometric equation 2sin(x) + 1 = 2.6635 for all solutions between [0, 360) degrees, we can follow these steps:

Step 1: Subtract 1 from both sides of the equation:
2sin(x) = 2.6635 - 1
2sin(x) = 1.6635

Step 2: Divide both sides of the equation by 2:
sin(x) = 1.6635 / 2
sin(x) = 0.83175

Step 3: In order to solve for x, we need to take the inverse sine (or arcsine) of both sides to cancel out the sine function. Since we are looking for solutions between [0, 360) degrees, we can use the principal value from the unit circle.

Using a scientific calculator or trigonometric tables, we find that the arcsin(0.83175) is approximately 56.28 degrees. However, there is one more solution to consider because sine is positive in both the first and second quadrants.

To find the other solution, we can use the fact that the sine function has a period of 360 degrees. Therefore, if sin(x) = 0.83175, then sin(x + 180) = sin(x).

Using the same approach, we can find that arcsin(0.83175) + 180 degrees is approximately 123.72 degrees.

So, the solutions for x are approximately 56.28 degrees and 123.72 degrees.