Answer to Question #287000 in Trigonometry for rose

Question

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.

Answer 1

Solution:

A.

Using Cosine formula,

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

Distance between galaxies is 70 light-years.

B.

Using Sine formula,

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