Use the Law of Sines to find the length of side c, to the nearest millimeter, if ∠A=72 degrees and side a=1,246 mm .
The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of the angle opposite that side is constant:
a/sin(A) = b/sin(B) = c/sin(C)
We are given that ∠A = 72 degrees and side a = 1,246 mm. We are looking to find the length of side c. We can set up the following equation using the Law of Sines:
1,246/sin(72) = c/sin(C)
To find the value of sin(72), we can use a calculator or a trigonometric table. sin(72) ≈ 0.9511.
Now, we can plug in the values we know into the equation:
1,246/0.9511 = c/sin(C)
1,309.75 ≈ c/sin(C)
To find the length of side c, we need to find sin(C). Since the sum of angles in a triangle is 180 degrees, we can find the measure of angle C:
∠A + ∠B + ∠C = 180
72 + ∠B + ∠C = 180
∠B + ∠C = 180 - 72
∠B + ∠C = 108
Therefore, we are looking to find sin(108). Using a calculator, sin(108) ≈ 0.9511.
Now we have:
1,309.75 ≈ c/0.9511
c ≈ 1,245.7
Therefore, side c is approximately 1,246 mm when rounded to the nearest millimeter.