SOLUTION: From a lookout tower A, a column of smoke is sighted due south. From a second tower B, 5.00 miles west of A, the smoke is observed in the direction South 63.0 degrees East. How far

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Question 718261: From a lookout tower A, a column of smoke is sighted due south. From a second tower B, 5.00 miles west of A, the smoke is observed in the direction South 63.0 degrees East. How far is the fire from B? from A?
Help please. I have no idea where to start with this.
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
It will probably help to draw a diagram.
  1. Draw a point and label it A.
  2. Since south is usually down on a map, draw a point somewhere below A and label it S (for smoke).
  3. Since west is usually to the left on maps, draw a point somewhere to the left of A and label it B.
  4. Draw in the sides of triangle ABS.
  5. Label the length of side AB as 5 since the towers are 5 miles apart.
  6. Label the length of side AS as x (since we do not yet know what it is).
  7. Label the length of side BS as y (since we do not yet know what it is).
  8. Label angle BAS as a right angle (since west and south are perpendicular).
  9. The tricky part is that the 63 degree angle is not in triangle ABS! Draw a segment south from B and label its endpoint as C. Angle CBS is the 63 degree angle. ("south 63 east" means an angle with one side south from B and the other side opening toward the east.).
  10. Since angle CBA is also a right angle and since angle CBS is 63, angle SBA is 27 degrees.
We now have a right triangle ABS with an acute angle, angle SBA, of 27 degrees. Side AB is adjacent to angle SBA, side AS is opposite to angle SBA and side BS is the hypotenuse.

To find x we need a trig ratio that involves x and the side we know, AB. Since x is opposite and AB is adjacent to the angle we know, we will use tan:

Multiplying both sides by 5:

I'll leave it up to you and your calculator to finish.

To find y we need a trig ratio that involves y and the side we know, AB. Since y is the hypotenuse and AB is adjacent to the angle we know, we will use cos:

Multiplying both sides by y:

Dividing by cos(27):

I'll leave it up to you and your calculator to finish.
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