Use the Law of Sines to solve the following problem. If ∠C
= 38° and side c
= 22mi., then how long is side a
to the nearest mile?
17 mi.
17 mi.
28 mi.
28 mi.
32 mi.
32 mi.
49 mi.
To solve this problem, we can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
In this case, we know that angle C is 38° and side c is 22 mi.
Using the Law of Sines, we can set up the following equation:
sin(A)/a = sin(C)/c
Plugging in the known values:
sin(A)/a = sin(38°)/22
To find side a, we want to isolate it on one side of the equation. We can do this by multiplying both sides by a:
sin(A) = a*sin(38°)/22
Now, to solve for a, we can take the inverse sine (or arcsine) of both sides:
A = arcsin(a*sin(38°)/22)
Using a calculator, we find that A is approximately 23.46°.
Now, we can use the fact that the sum of the angles in a triangle is 180° to find angle B:
B = 180° - A - C
B = 180° - 23.46° - 38°
B = 118.54°
Finally, we can use the Law of Sines again to find side a:
sin(A)/a = sin(B)/b
sin(23.46°)/a = sin(118.54°)/b
To find side a, we can rearrange the equation and solve for it:
a = b*sin(23.46°)/sin(118.54°)
Plugging in the known values:
a = 22*sin(23.46°)/sin(118.54°)
Using a calculator, we find that a is approximately 17 mi.
Therefore, the length of side a to the nearest mile is 17 mi.