For sin(θ) = -0.4321, 0 ≤ θ ≤ 360°. (8)

How many solutions are possible?
In which quadrants would you find the solutions?
Determine the reference angle for this equation to the nearest degree.
Determine all the solutions to the equation to the nearest degree.

sketch a sine curve. look at where it can be -.43.

We will be happy to check your work.

To determine the number of solutions for the equation sin(θ) = -0.4321, we need to consider the range of values for θ and the periodic nature of the sine function.

1. The equation sin(θ) = -0.4321 has an infinite number of solutions since the sine function repeats every 360 degrees.

Next, let's find the quadrants in which the solutions can be found:

2. In the unit circle, sine is positive in the first and second quadrants, and negative in the third and fourth quadrants. Since sin(θ) = -0.4321 is negative, the solutions will be located in the third and fourth quadrants.

Now, let's determine the reference angle for this equation:

3. Since we have a negative value for the sine, we can consider the reference angle as positive. To find the reference angle, we can use the inverse sine function (also known as arcsine), which is denoted as sin^(-1) or asin. So, sin^(-1)(0.4321) ≈ 25.99 degrees. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis.

Finally, let's determine all the solutions to the equation:

4. Since the reference angle is 25.99 degrees, we can determine the solutions by considering the third and fourth quadrants which have a sine value of -0.4321.

In the third quadrant, the angle will be 180 degrees plus the reference angle, so the solution in the third quadrant is approximately 180 + 25.99 ≈ 205.99 degrees.

In the fourth quadrant, the angle will be 360 degrees minus the reference angle, so the solution in the fourth quadrant is approximately 360 - 25.99 ≈ 334.01 degrees.

Therefore, the solutions to the equation sin(θ) = -0.4321, with 0 ≤ θ ≤ 360°, are approximately 205.99 degrees and 334.01 degrees.