Chapter D, Problem 84E

In order to find the distance |AB| across a small inlet, a point C was located as in the figure and the following measurements were recorded:

$\begin{array}{lll}\angle C=103°\hfill & \left|AC\right|=820\text{m}\hfill & \left|BC\right|=910\text{m}\hfill \end{array}$

Use the Law of Cosines from Exercise 83 to find the required distance.

83. Prove the Law of Cosines: If a triangle has sides with lengths a, b, and c, and θ is the angle between the sides with lengths a and b, then

$c^2=a^2+b^2-2ab\cos \theta$

[Hint: Introduce a coordinate system so that θ is in standard position, as in the figure. Express x and y in terms of θ and then use the distance formula to compute c.]