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Difficulty: MediumLaw of Sines and Law of Cosines

A sailboat is traveling along a straight path. From point AA, a lighthouse LL is observed at an angle of 35∘35^\circ relative to the line of travel. After the boat travels 400400 meters directly along the path to reach point CC, the angle to the lighthouse relative to the continuing line of travel increases to 65∘65^\circ. Which of the following expressions represents the distance, in meters, from point CC to the lighthouse LL?

  1. 800sin⁡(35∘)800 \sin(35^\circ)Answer
  2. B
    400sin⁡(35∘)400 \sin(35^\circ)
  3. C
    400sin⁡(35∘)sin⁡(65∘)\frac{400 \sin(35^\circ)}{\sin(65^\circ)}
  4. D
    400sin⁡(65∘)sin⁡(35∘)\frac{400 \sin(65^\circ)}{\sin(35^\circ)}
  5. E
    800sin⁡(65∘)800 \sin(65^\circ)

Answer

800sin⁡(35∘)800 \sin(35^\circ) meters
To find the distance from CC to the lighthouse LL, analyze △ALC\triangle ALC. The given angle at AA is 35∘35^\circ. Point CC is along the straight line of travel, so the interior angle ∠LCA=180∘−65∘=115∘\angle LCA = 180^\circ - 65^\circ = 115^\circ. The top angle ∠ALC=180∘−(35∘+115∘)=30∘\angle ALC = 180^\circ - (35^\circ + 115^\circ) = 30^\circ. By the Law of Sines, CLsin⁡(35∘)=400sin⁡(30∘)\frac{CL}{\sin(35^\circ)} = \frac{400}{\sin(30^\circ)}. Since sin⁡(30∘)=0.5\sin(30^\circ) = 0.5, solving for CLCL gives CL=400sin⁡(35∘)0.5=800sin⁡(35∘)CL = \frac{400 \sin(35^\circ)}{0.5} = 800 \sin(35^\circ).

Step-by-Step Solution

1
Determine the interior angles of △ALC\triangle ALC
Angle ∠LAC=35∘\angle LAC = 35^\circ. The supplementary interior angle at CC is ∠LCA=180∘−65∘=115∘\angle LCA = 180^\circ - 65^\circ = 115^\circ. The third interior angle ∠ALC=180∘−(35∘+115∘)=30∘\angle ALC = 180^\circ - (35^\circ + 115^\circ) = 30^\circ.
The interior angle and exterior angle along a straight path sum to 180∘180^\circ, and the interior angles of a triangle sum to 180∘180^\circ.
2
Apply the Law of Sines to find distance CLCL
CLsin⁡(35∘)=ACsin⁡(30∘)  ⟹  CLsin⁡(35∘)=400sin⁡(30∘)\frac{CL}{\sin(35^\circ)} = \frac{AC}{\sin(30^\circ)} \implies \frac{CL}{\sin(35^\circ)} = \frac{400}{\sin(30^\circ)}
The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant in any triangle.
3
Simplify the expression using sin⁡(30∘)=0.5\sin(30^\circ) = 0.5
CL=400sin⁡(35∘)0.5=800sin⁡(35∘)CL = \frac{400 \sin(35^\circ)}{0.5} = 800 \sin(35^\circ)
Dividing 400400 by 0.50.5 yields 800800.

Key Concept

Law of Sines
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Topics

Law of Sines and Law of Cosines

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