Answer :

Answer:

1. [tex]D\approx 43.2\°[/tex]

2. [tex]JH\approx 30.9\text{ ft}[/tex]

Step-by-step explanation:

1. We can solve for the length of side DF using the Law of Cosines:

[tex]c^2=a^2+b^2-2ab\cos(C)[/tex]

where:

  • [tex]c[/tex] is the side opposite angle [tex]C[/tex].

Plugging in the given values, we get:

[tex](DF)^2=19^2+31^2-2(19)(31)\cos(112\°)[/tex]

↓ taking the square root of both sides

[tex]DF = \sqrt{19^2+31^2-2(19)(31)\cos(112\°)}[/tex]

evaluating using a calculator

[tex]DF\approx 41.9915\text{ ft}[/tex]

Next, we can solve for the measure of angle D using the Law of Sines:

[tex]\dfrac{\sin(A)}{a}=\dfrac{\sin(B)}{b}[/tex]

where:

  • [[tex]A[/tex] and [tex]a[/tex]] and [[tex]B[/tex] and [tex]b[/tex]] are each a pair of an angle and its opposite side.

Plugging in the known values, we get:

[tex]\dfrac{\sin(D)}{31} = \dfrac{\sin(112\°)}{41.9915}[/tex]

multiplying both sides by 31

[tex]\sin(D) = \dfrac{31\sin(112\°)}{41.9915}[/tex]

↓ taking the inverse sine of both sides

[tex]D = \sin^{-1}\!\left(\dfrac{31\sin(112\°)}{41.9915}\right)[/tex]

evaluating using a calculator

[tex]\boxed{D\approx 43.2\°}[/tex]

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2. We can solve for the length of side JH using the Law of Sines.

First, we have to solve for the measure of angle H using our knowledge that the measures of the interior angles of a triangle add to 180°:

[tex]25\°+43\°+H=180\°[/tex]

↓ subtracting [tex](25\°+43\°)[/tex] from both sides

[tex]H = 112\°[/tex]

Next, we can plug the known values into the Law of Sines:

[tex]\dfrac{\sin(43\°)}{JH}=\dfrac{\sin(112\°)}{42}[/tex]

↓ taking the reciprocal of both sides

[tex]\dfrac{JH}{\sin(43\°)}=\dfrac{42}{\sin(112\°)}[/tex]

multiplying both sides by [tex]\sin(43\°)[/tex]

[tex]JH=\dfrac{42\sin(43\°)}{\sin(112\°)}[/tex]

evaluating using a calculator

[tex]\boxed{JH\approx 30.9\text{ ft}}[/tex]