Question

an astronomer is 450 light years (l.y.) from star a and 400 l.y. from star b, as shown. star a is 90 l.y. from star b. if the astronomer’s telescope is currently pointed at star a, how many degrees must she rotate her telescope to see star b? do not round during your calculations. round your final answer to the nearest degree.

Explanation:

Step1: Identify the triangle sides

We have a triangle with sides: \( a = 90 \) (distance between star A and B), \( b = 450 \) (distance from astronomer to star A), \( c = 400 \) (distance from astronomer to star B). We need to find the angle \( \theta \) opposite side \( a \), using the Law of Cosines.

The Law of Cosines formula is: \( \cos\theta=\frac{b^{2}+c^{2}-a^{2}}{2bc} \)

Step2: Substitute the values

Substitute \( a = 90 \), \( b = 450 \), \( c = 400 \) into the formula:

\( \cos\theta=\frac{450^{2}+400^{2}-90^{2}}{2\times450\times400} \)

First, calculate the numerator:

\( 450^{2}=202500 \), \( 400^{2}=160000 \), \( 90^{2}=8100 \)

\( 202500 + 160000- 8100=202500 + 151900 = 354400 \)

Denominator: \( 2\times450\times400 = 360000 \)

So, \( \cos\theta=\frac{354400}{360000}\approx0.984444 \)

Step3: Find the angle

To find \( \theta \), take the inverse cosine (arccos) of \( 0.984444 \):

\( \theta=\arccos(0.984444) \)

Using a calculator, \( \theta\approx9.9^{\circ}\approx10^{\circ} \) (rounded to the nearest degree)

Answer:

\( 10 \)