Question

1. viewed from earth, two stars form an angle of 76.04 °. star a is 23.30 light - years from earth, and star b is 34.76 light - years from earth. sketch a diagram modeling this situation and find how many light - years the stars are from each other. (3 points)

Explanation:

Step1: Identify the Law of Cosines

We use the Law of Cosines for triangle \(ABC\) (Earth = \(A\), Star A = \(B\), Star B = \(C\)):
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$
Here, \(a=23.30\) ly, \(b=34.76\) ly, \(C=76.04^\circ\), \(c\) = distance between stars.

Step2: Calculate \(2ab\cos(C)\)

Compute the product and cosine term:
$$2 \times 23.30 \times 34.76 \times \cos(76.04^\circ) \approx 2 \times 23.30 \times 34.76 \times 0.2417 \approx 393.5$$

Step3: Compute \(a^2 + b^2\)

Calculate the sum of squares:
$$23.30^2 + 34.76^2 = 542.89 + 1208.2576 = 1751.1476$$

Step4: Solve for \(c\)

Subtract and take the square root:
$$c = \sqrt{1751.1476 - 393.5} = \sqrt{1357.6476} \approx 34.1$$

Answer:

34.1 light-years