THE LAW OF SINES
Example 3:
For the triangle with measures a = 15 inches, b = 25 inches, and A = 85o, find the remaining sides
and angles. Round the final solutions to two decimal places. HINT: Make a sketch!!!
Solve for Angle B:
Using the Law of Sines,
, we'll write
However, the result contradicts the fact that the numeric value of sin B is always between
and including -1 and 1. Your calculator will give you an ERROR message.
So, no triangle can be formed. Below is a more accurate picture of the given information:
Example 4:
Two towers on the side of a highway are 30 miles apart with tower A being due west of tower B. A
fire is spotted from both of the towers with a bearing of E 76o N from tower A to the fire and a
bearing of W 56o N from tower B to the fire. Find the distance d (shortest distance!!!) from the
highway to the Fire at point F. Round all solutions to two decimal places.
NOTE: The shortest distance between a point and a given line always makes a 90o angle with the
line.
We can use the Law of Sines,
. Since we need one angle and its opposite
side, let's find angle at the point F, which is opposite line segment AB which is 30 miles long.
Given a bearing of E 76o N from tower A to the fire, we know that angle A is 90o - 76o = 14o.
Given a bearing of W 56o N from tower B to the fire, we know that angle B is 90o - 56o = 34o.
Then we can calculate angle F as follows.
Now we can find line segment BF, let's call it a for short.
DO NOT CALCULATE the individual numeric values of the trigonometric
ratios, instead use the EXACT value as shown above.
Using the sine ratio to find distance d, we get
.
Thus, the shortest distance from the highway to the fire is approximately 5.46 miles.