An aircraft is timetabled to travel from A to B. Due to bad weather it flies from A to C then from C to B. Where AC and CB make angles of 27 and 66 degrees respectively with AB. If AC =220km find AB
240.5
To find the length of AB, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the relationship between them is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label AB as c, AC as a, and CB as b. We are given AC as 220 km, and the angles between AB and AC (27 degrees) and AB and CB (66 degrees).
We can first find the length of CB using the law of cosines by rearranging the formula:
CB^2 = AC^2 + AB^2 - 2AC * AB * cos(27)
Since AC = 220 km, we can substitute this value in and solve for CB^2:
CB^2 = (220)^2 + AB^2 - 2(220)(AB) * cos(27)
Similarly, using the information for angle C = 66 degrees:
AB^2 = AC^2 + CB^2 - 2AC * CB * cos(66)
Substituting the known values:
AB^2 = (220)^2 + CB^2 - 2(220)(CB) * cos(66)
Now we have two equations with two variables, CB and AB. We can solve this system of equations simultaneously to find AB.
1. CB^2 = (220)^2 + AB^2 - 2(220)(AB) * cos(27)
2. AB^2 = (220)^2 + CB^2 - 2(220)(CB) * cos(66)
Using a numerical method or solving manually, we can find the value of AB.
calculate the third angle in the triangle. Then, law of sines will work easily to get AB
Answer
Angle C = 87°(calculated)
AB=C
AC=220=b
Sine laws c\sin87=220\66
c=240.5