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Difficulty: MediumLaw of Sines and Law of Cosines

Two radar stations, AA and BB, are located 1212 miles apart along a straight coastline. Both stations track a ship offshore at point CC. The angle formed by the coastline ABAB and the line of sight from station AA to the ship (∠BAC\angle BAC) measures 42∘42^\circ, and the angle formed by the coastline ABAB and the line of sight from station BB to the ship (∠ABC\angle ABC) measures 78∘78^\circ. Which of the following expressions represents the distance, in miles, from station AA to the ship?

  1. A
    12sin⁡(42∘)sin⁡(60∘)\frac{12 \sin(42^\circ)}{\sin(60^\circ)}
  2. 12sin⁡(78∘)sin⁡(60∘)\frac{12 \sin(78^\circ)}{\sin(60^\circ)}Answer
  3. C
    12sin⁡(60∘)sin⁡(78∘)\frac{12 \sin(60^\circ)}{\sin(78^\circ)}
  4. D
    12sin⁡(78∘)sin⁡(42∘)\frac{12 \sin(78^\circ)}{\sin(42^\circ)}
  5. E
    12sin⁡(42∘)sin⁡(78∘)\frac{12 \sin(42^\circ)}{\sin(78^\circ)}

Answer

The distance, in miles, from station A to the ship is given by 12sin⁡(78∘)sin⁡(60∘)\frac{12 \sin(78^\circ)}{\sin(60^\circ)}.
The sum of angles in △ABC\triangle ABC is 180∘180^\circ, so ∠C=180∘−42∘−78∘=60∘\angle C = 180^\circ - 42^\circ - 78^\circ = 60^\circ. The distance from station A to the ship corresponds to side length ACAC, which lies opposite ∠B=78∘\angle B = 78^\circ. Applying the Law of Sines yields ACsin⁡(78∘)=12sin⁡(60∘)\frac{AC}{\sin(78^\circ)} = \frac{12}{\sin(60^\circ)}, which simplifies to AC=12sin⁡(78∘)sin⁡(60∘)AC = \frac{12 \sin(78^\circ)}{\sin(60^\circ)}.

Step-by-Step Solution

1
Calculate the measure of the third angle ∠ACB\angle ACB in △ABC\triangle ABC.
∠ACB=180∘−(42∘+78∘)=60∘\angle ACB = 180^\circ - (42^\circ + 78^\circ) = 60^\circ
The interior angles of any triangle must sum to 180∘180^\circ.
2
Set up the Law of Sines relating the known side AB=12AB = 12 and its opposite angle ∠ACB=60∘\angle ACB = 60^\circ to the unknown side AC=bAC = b and its opposite angle ∠ABC=78∘\angle ABC = 78^\circ.
ACsin⁡(78∘)=12sin⁡(60∘)\frac{AC}{\sin(78^\circ)} = \frac{12}{\sin(60^\circ)}
The Law of Sines states that asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
3
Solve the equation for ACAC.
AC=12sin⁡(78∘)sin⁡(60∘)AC = \frac{12 \sin(78^\circ)}{\sin(60^\circ)}
Multiply both sides of the equation by sin⁡(78∘)\sin(78^\circ).

Key Concept

Law of Sines
Estimated Time:1m 15s
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Topics

Law of Sines and Law of Cosines

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