Two observation posts are 12km apart at A and B. From A, a third observation post C is located such that angle CAB is 42 degrees while CBA is 67 degrees. Find the distance of C from both A and B.

To find the distance of post C from both A and B, we can use the law of sines.

Let's denote the distance of post C from A as AC and the distance of post C from B as BC.

According to the law of sines, the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Thus, we can write:

AC / sin(CBA) = AB / sin(CAB)

Substituting the given values:

AC / sin(67°) = 12 km / sin(42°)

To find the value of AC, we need to rearrange the equation and solve for it:

AC = (12 km * sin(67°)) / sin(42°)

Using a calculator, we can evaluate the right side of the equation:

AC ≈ (12 km * 0.921) / 0.669

AC ≈ 16.5 km

Similarly, we can find the distance of post C from B:

BC = (12 km * sin(42°)) / sin(67°)

Using a calculator:

BC ≈ (12 km * 0.669) / 0.921

BC ≈ 8.7 km

Therefore, the distance of post C from post A is approximately 16.5 km, and the distance of post C from post B is approximately 8.7 km.

To find the distance of point C from points A and B, we can use the Law of Sines. The Law of Sines states that for any triangle ABC, the ratio of the length of a side to the sine of the opposite angle is constant.

In this case, we are given the lengths of sides AB (12km) and BC (unknown) and the measures of angles CAB (42 degrees) and CBA (67 degrees). Let's assign the length of side AC as 'x' km.

From the Law of Sines, we have:

sin(angle CAB) / side AC = sin(angle CBA) / side BC

Substituting the given values:

sin(42 degrees) / x = sin(67 degrees) / 12km

Now we can solve for x, the distance of point C from point A:

x = sin(42 degrees) * 12km / sin(67 degrees)

Using a calculator, we find:

x ≈ 8.702km

Therefore, the distance of point C from point A is approximately 8.702km. To find the distance of point C from point B, we can simply subtract this value from the distance between A and B:

Distance of C from B = Distance of A from B - Distance of C from A
= 12km - 8.702km
≈ 3.298km

Therefore, the distance of point C from point B is approximately 3.298km.

Did u try drawing this out? :)

If not try that.