Answer :

To solve triangle ABC with the given information, we have the following details:

  • Angle B = 35° 00'
  • Side a = 38.5
  • Side b = 30.9

We need to find the missing parts of the triangle, namely angle A, angle C, and side c.

Step-by-step Solution:

  1. Find Angle A using the Law of Sines:

    We can use the Law of Sines, which states:
    [tex]\frac{a}{\sin A} = \frac{b}{\sin B}[/tex]

    Plug in the known values:
    [tex]\frac{38.5}{\sin A} = \frac{30.9}{\sin 35°}[/tex]

    First, calculate [tex]\sin 35°[/tex]:
    [tex]\sin 35° \approx 0.5736[/tex]

    Substitute [tex]\sin 35°[/tex] into the equation:
    [tex]\frac{38.5}{\sin A} = \frac{30.9}{0.5736}[/tex]

    [tex]\frac{38.5}{\sin A} = 53.87[/tex]

    Solve for [tex]\sin A[/tex]:
    [tex]\sin A = \frac{38.5}{53.87}[/tex]

    [tex]\sin A \approx 0.7148[/tex]

    Find angle A:
    [tex]A \approx \sin^{-1}(0.7148)[/tex]

    [tex]A \approx 45.60°[/tex]

  2. Find Angle C:

    Since the sum of angles in a triangle is 180°, we can find angle C:
    [tex]C = 180° - B - A[/tex]

    [tex]C = 180° - 35° - 45.60°[/tex]

    [tex]C \approx 99.40°[/tex]

  3. Find Side c using the Law of Sines:

    Use the Law of Sines again:
    [tex]\frac{c}{\sin C} = \frac{a}{\sin A}[/tex]

    Plug in the known values:
    [tex]\frac{c}{\sin 99.40°} = \frac{38.5}{\sin 45.60°}[/tex]

    Calculate [tex]\sin 99.40°[/tex] and [tex]\sin 45.60°[/tex]:
    [tex]\sin 99.40° \approx 0.9848[/tex]
    [tex]\sin 45.60° \approx 0.7150[/tex]

    Substitute into the equation:
    [tex]\frac{c}{0.9848} = \frac{38.5}{0.7150}[/tex]

    [tex]c \approx 53.02[/tex]

Now we have all the elements of triangle ABC:

  • Angle A [tex]\approx 45.60°[/tex]
  • Angle B [tex]= 35° 00'[/tex]
  • Angle C [tex]\approx 99.40°[/tex]
  • Side a [tex]= 38.5[/tex]
  • Side b [tex]= 30.9[/tex]
  • Side c [tex]\approx 53.02[/tex]

This solution completes the triangle ABC using the provided information.

Thanks for taking the time to read Solve each triangle ABC that exists given the following B 35 00 a 38 5 b 30 9. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada