Sure, let's solve the equation [tex]\( 1.33 \sin 25.0^{\circ} = 1.50 \sin \theta \)[/tex] step-by-step.
Step 1: Find [tex]\(\sin 25.0^{\circ}\)[/tex]
First, we need the value of [tex]\(\sin 25.0^{\circ}\)[/tex]. We know that:
[tex]\[
\sin 25.0^{\circ} \approx 0.4226
\][/tex]
Step 2: Write the equation relating [tex]\(\theta\)[/tex] and [tex]\(\sin 25.0^{\circ}\)[/tex]
Given the equation:
[tex]\[
1.33 \sin 25.0^{\circ} = 1.50 \sin \theta
\][/tex]
Step 3: Substitute [tex]\(\sin 25.0^{\circ}\)[/tex] with its value
Plug in the value of [tex]\(\sin 25.0^{\circ}\)[/tex]:
[tex]\[
1.33 \cdot 0.4226 = 1.50 \sin \theta
\][/tex]
Step 4: Simplify the left-hand side
Calculate [tex]\(1.33 \cdot 0.4226\)[/tex]:
[tex]\[
0.5610 = 1.50 \sin \theta
\][/tex]
Step 5: Isolate [tex]\(\sin \theta\)[/tex]
To find [tex]\(\sin \theta\)[/tex], divide both sides of the equation by 1.50:
[tex]\[
\sin \theta = \frac{0.5610}{1.50}
\][/tex]
[tex]\[
\sin \theta \approx 0.3747
\][/tex]
Step 6: Determine the angle [tex]\(\theta\)[/tex]
Finally, we need to find [tex]\(\theta\)[/tex] such that [tex]\(\sin \theta \approx 0.3747\)[/tex]. We use the inverse sine function ([tex]\(\sin^{-1}\)[/tex] or [tex]\(\arcsin\)[/tex]):
[tex]\[
\theta = \arcsin(0.3747)
\][/tex]
[tex]\[
\theta \approx 22.0^{\circ}
\][/tex]
So, the calculated [tex]\(\sin \theta\)[/tex] is approximately [tex]\(0.3747\)[/tex] and [tex]\(\theta\)[/tex] is approximately [tex]\(22.0^{\circ}\)[/tex].
