Use the Law of Sines to solve the triangle. (Round your answers to two decimal places.)
A = 76°, a = 38, b = 29
B =
C =
c =
SinA/a = SinB/b
Sin76/38 = SinB/29
re-arrange and solve for B : )
To solve the triangle using the Law of Sines, we can start by understanding the Law of Sines itself. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. In mathematical terms, it can be written as:
a/sinA = b/sinB = c/sinC
Here, a, b, and c represent the lengths of the sides of the triangle, while A, B, and C represent the opposite angles.
To find angle B, we can use the given values of A and a. Rearranging the Law of Sines equation, we have:
sinB = b * sinA / a
Plugging in the given values, we get:
sinB = 29 * sin(76°) / 38
To find angle B, we can take the inverse sine (sin⁻¹) of both sides:
B = sin⁻¹(29 * sin(76°) / 38)
Calculating this value in a calculator, B is approximately 46.70 degrees (rounded to two decimal places).
Next, to find angle C, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Since we already know angles A and B, we can calculate angle C by subtracting their sum from 180 degrees:
C = 180° - A - B
= 180° - 76° - 46.70°
Calculating this value, C is approximately 57.30 degrees (rounded to two decimal places).
Finally, to find the remaining side c, we can use the Law of Sines again:
c/sinC = a/sinA
Rearranging the equation, we get:
c = a * sinC / sinA
Plugging in the given values, we have:
c = 38 * sin(57.30°) / sin(76°)
Calculating this value, c is approximately 44.39 units (rounded to two decimal places).