Question

Use the law of sines to find the length of side c, to the nearest millimeter, if angle A=72 degrees and a side a=1,246mm.

385 mm
366 mm
3,835 mm
405 mm

Answer 1

To use the law of sines, we need to find another angle or side in the triangle. In this case, we have angle A and side a. We can use the law of sines formula:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Since we only have one angle (A) and one side (a), we need either another angle or side to continue. If no additional information is provided, we can take a theoretical approach: calculate the maximum possible side length \(c\) assuming \(B\) and \(C\) are configured such that side \(c\) gets maximized.

For triangle calculations, let’s pick \(B = 90^\circ\) for simplicity (right triangle) since no other angles are given.

This means:

- \(A = 72^\circ\)
- \(B = 90^\circ\)
- \(C = 180^\circ - A - B = 180 - 72 - 90 = 18^\circ\)

Now we will find \(c\) using the law of sines:

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

\[
\frac{1246 \, \text{mm}}{\sin(72^\circ)} = \frac{c}{\sin(18^\circ)}
\]

Calculating \(\sin(72^\circ)\) and \(\sin(18^\circ)\):

- \(\sin(72^\circ) \approx 0.9511\)
- \(\sin(18^\circ) \approx 0.3090\)

Now, substituting these values into the equation:

\[
\frac{1246}{0.9511} = \frac{c}{0.3090}
\]

Calculating the left side:

\[
\frac{1246}{0.9511} \approx 1310.81
\]

So we have:

\[
1310.81 = \frac{c}{0.3090}
\]

Now, solve for \(c\):

\[
c = 1310.81 \times 0.3090
\]

\[
c \approx 405.0 \, \text{mm}
\]

Rounding to the nearest millimeter, we get:

\[
c \approx 405 \, \text{mm}
\]

Thus, the length of side \(c\) is **405 mm**.