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]
