if the area of △ABC=92.5 Km^2, ∠C=73 degrees, abd b=16 km, then what is the length of a to the nearest km if the sin is 15.3008
To find the length of side a in triangle ABC, we can use the sine rule.
The sine rule states that in any triangle ABC with sides a, b, c and opposite angles A, B, C:
a/sin(A) = b/sin(B) = c/sin(C)
Given that ∠C = 73 degrees and b = 16 km, we know:
c/sin(73) = 16/sin(B)
Let's solve for sin(B):
sin(B) = (16 * sin(73)) / c
Now, we are given that sin(B) = 15.3008, so we can substitute this value in the equation:
15.3008 = (16 * sin(73)) / c
To find c, we rearrange the equation:
c = (16 * sin(73)) / 15.3008
Now, we know that the area of the triangle is 92.5 km². The area of a triangle can be calculated using the formula:
Area = (1/2) * a * b * sin(C)
Substituting the given values:
92.5 = (1/2) * a * 16 * sin(73)
To find the length of side a, rearrange the equation:
a = (92.5 * 2) / (16 * sin(73))
Calculating this expression, we find:
a ≈ 17.876
Rounding to the nearest kilometer, the length of side a is 18 km.