To find all angles [tex]\(\theta\)[/tex] between [tex]\(0^\circ\)[/tex] and [tex]\(180^\circ\)[/tex] that satisfy the equation [tex]\(\sin(\theta) = \frac{5}{8}\)[/tex], follow these steps:
1. Background Concepts:
- The sine function [tex]\( \sin(\theta) \)[/tex] gives the ratio of the length of the opposite side to the hypotenuse in a right triangle.
- The range of the sine function is between -1 and 1.
- The inverse sine function, denoted as [tex]\(\sin^{-1}\)[/tex] or [tex]\(\text{arcsin}\)[/tex], is used to find an angle when the sine value is known.
2. Initial Calculation:
- Start by finding the angle [tex]\(\theta\)[/tex] whose sine is [tex]\(\frac{5}{8}\)[/tex].
- Use the inverse sine (arcsin) function to find this angle in radians.
- Convert the result from radians to degrees.
3. Converting Angle:
- Once you find the initial angle [tex]\(\theta\)[/tex], remember that sine is positive in both the first quadrant [tex]\((0^\circ \text{ to } 90^\circ)\)[/tex] and the second quadrant [tex]\((90^\circ \text{ to } 180^\circ)\)[/tex].
4. Second Solution:
- For any angle [tex]\(\theta\)[/tex] in the first quadrant, there is a corresponding angle in the second quadrant, which is [tex]\(180^\circ - \theta\)[/tex].
5. Rounding:
- Round the results to one decimal place for precision.
Now, let’s outline the numerical results:
- Using the inverse sine function and converting to degrees, the first angle [tex]\(\theta \approx 38.7^\circ\)[/tex].
- The corresponding angle in the second quadrant is [tex]\(180^\circ - 38.7^\circ\)[/tex], which is approximately [tex]\(141.3^\circ\)[/tex].
Thus, the two angles between [tex]\(0^\circ\)[/tex] and [tex]\(180^\circ\)[/tex] that satisfy [tex]\(\sin(\theta) = \frac{5}{8}\)[/tex] are:
[tex]\(\theta = 38.7^\circ\)[/tex] and [tex]\(\theta = 141.3^\circ\)[/tex].
These are the solutions, rounded to one decimal place.
