Final answer:
To find the angle θ when sin θ = 0.819, use the inverse sine function to get θ ≈ 54.5 degrees. This calculation uses the principal value from the calculator, which lies in the first quadrant.
Step-by-step explanation:
If sin θ = 0.819, to find the angle θ, we use the inverse sine function, often denoted as sin-1 or arcsin. Since the sine function can only produce values between -1 and 1, and given that sin θ = 0.819 is within this range, we can proceed to use a calculator to evaluate this expression and find the angle in degrees.
The angle θ is the arcsin of 0.819. When calculated, this is approximately θ = sin-1(0.819) ≈ 54.5°. It's important to note that this will give us the principal value of the angle, which lies within the range of [-90°, 90°]. Depending on the context of the problem, there may be other angles that have the same sine value, which would be in the second quadrant (180° - θ).
Discussion about the value of sin not being greater than 1 clarifies that the given value of 0.819 is a valid input for the inverse sine function, thus ensuring that our calculation for θ is meaningful and correct.