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(05.05 LC)

If [tex]\angle P[/tex] measures [tex]27^{\circ}[/tex], [tex]\angle R[/tex] measures [tex]135^{\circ}[/tex], and [tex]p[/tex] equals [tex]9.5[/tex], write an equation to find the length of [tex]r[/tex] using only the Law of Sines.

[tex]\frac{\sin 27^{\circ}}{r} = \frac{\sin 135^{\circ}}{9.5}[/tex]



Answer :

GinnyAnswer
To find the length of [tex]\( r \)[/tex] using the Law of Sines, we can use the formula:

[tex]\[ \frac{\sin \angle P}{p} = \frac{\sin \angle R}{r} \][/tex]

Given:
- [tex]\(\angle P = 27^\circ\)[/tex]
- [tex]\(\angle R = 135^\circ\)[/tex]
- [tex]\( p = 9.5 \)[/tex]

We'll first need to express the sine values of the given angles.

To execute the steps:

1. Convert angles from degrees to radians:
- [tex]\(\angle P\)[/tex] in radians: [tex]\( \angle P_{\text{rad}} \approx 0.4712 \)[/tex]
- [tex]\(\angle R\)[/tex] in radians: [tex]\( \angle R_{\text{rad}} \approx 2.3562 \)[/tex]

2. Calculate the sine values:
- [tex]\( \sin(27^\circ) \approx 0.454 \)[/tex]
- [tex]\( \sin(135^\circ) \approx 0.707 \)[/tex]

3. Use the Law of Sines to create the equation:
[tex]\[ \frac{\sin(27^\circ)}{9.5} = \frac{\sin(135^\circ)}{r} \][/tex]

4. Rearrange to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{9.5 \cdot \sin(135^\circ)}{\sin(27^\circ)} \][/tex]

5. Substitute the sine values into the equation:
[tex]\[ r = \frac{9.5 \cdot 0.707}{0.454} \][/tex]

6. Calculate [tex]\( r \)[/tex]:
[tex]\[ r \approx 14.7966 \][/tex]

Therefore, the length of [tex]\( r \)[/tex] is approximately [tex]\( 14.7966 \)[/tex].

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