Question

a lighthouse views two boats from the shore, one 24 m north, and the other 20 m east. the boats are 28 m apart. what is the angle between the lighthouses lines of sight?
lighthouse’s angle = ?°
round your answer to the nearest hundredth.

Explanation:

Step1: Identify the triangle sides

We have a triangle with sides \( a = 24 \), \( b = 20 \), and \( c = 28 \). We need to find the angle \( C \) (at the lighthouse) using the Law of Cosines. The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \).

Step2: Rearrange the formula for \( \cos(C) \)

Rearranging the Law of Cosines to solve for \( \cos(C) \), we get \( \cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab} \).

Step3: Substitute the values

Substitute \( a = 24 \), \( b = 20 \), and \( c = 28 \) into the formula:

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Step4: Find the angle \( C \)

To find \( C \), we take the inverse cosine (arccos) of \( 0.2 \). So \( C=\arccos(0.2) \). Using a calculator, \( \arccos(0.2)\approx78.46^{\circ} \).

Answer:

\( 78.46 \)