Answer to Question #287000 in Trigonometry for rose

Question #287000

APPLICATIONS OF TRIGONOMETRY

  1. An astronomer finds that the distance from planet A to galaxy B is 30 light years and the distance from A to galaxy C is 80 light years. the angle between distance AB and distance AC is measured to be 60 degree.

A. how far apart are the galaxies? (distance of galaxy B to galaxy C); and

B. Find the angle between distance AB to distance BC.

*show complete solution.


1
Expert's answer
2022-01-13T10:55:58-0500

Solution:


A.

Using Cosine formula,

"BC^2=AB^2+AC^2-2\\ AB.AC\\cos A\n\\\\ \\Rightarrow BC^2=30^2+80^2-2(30)(80)\\cos 60\\degree\n\\\\\\Rightarrow BC^2=7300-2400\n\\\\ \\Rightarrow BC=\\sqrt{4900}\n\\\\ \\Rightarrow BC=70\\ ly"

Distance between galaxies is 70 light-years.

B.

Using Sine formula,

"\\dfrac{BC}{\\sin A}=\\dfrac{AC}{\\sin B}\n\\\\ \\Rightarrow \\dfrac{70}{\\sin 60\\degree}=\\dfrac{80}{\\sin B}\n\\\\\\Rightarrow \\dfrac{70}{\\sqrt3\/2}=\\dfrac{80}{\\sin B}\n\\\\ \\Rightarrow \\sin B=0.989\n\\\\ \\Rightarrow B=81.49\\degree"


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