Question

Island A is 220 miles from island B. A ship captain travels 310 miles from island A and then finds that he is off course and 170 miles from island B. What angle, in degrees, must he turn through to head straight for island B? Round the answer to two decimal places. (Hint: Be careful to properly identify which angle is the turning angle.)
Group of answer choices

43.38°

136.62°

46.62°

None of the other answers are correct

93.23°

Answer 1

I drew a little diagram to help you visualize the question, where the points represent the location of island A, B, and the Ship, and angle x is what we must find; I am still not entirely sure exactly how to go about finding the answer.
I suspect the solution has something to do with sin or cosine properties, or perhaps the ratio between angles and side lengths? I'll try to investigate a bit further

Answer 2

As @snowflake0531 helpfully pointed out to me, you can use something called the "law of cosines" or "cosine rule" in this situation; when you know all the side lengths of a triangle, you can use this rule to calculate the angle measures.
The details are described here https://byjus.com/maths/cosine-rule/#:~:text=In%20trigonometry%2C%20the%20Cosine%20Rule%20says%20that%20the,also%20called%20law%20of%20cosines%20or%20Cosine%20Formula.
In my next comment, I'll use this rule to help us solve this question

Answer 3

The cosine law states that a triangle with side a opposite of angle x, side b opposite of angle y, and side c opposite of angle z follows the following proportions:
cos x = (b^2 + c^2 -a^2)/2bc
cos y = (a^2 + c^2 -b^2)/2ac
cos z = (a^2 + b^2 – c^2)/2ab
In our case, angle "x" is opposite of the side length measuring 220 miles, while the other two sides, measuring 310 and 170, are adjacent. We can plug these into the equation in order to solve for x.
cos x = (310^2 + 170^2 - 220^2)/2*(310*170)
which evaluates to
cos x = (96100 + 28900 - 48400)/105400
which simplifies to
cos x = 0.72675
Taking the inverse cosine of this value gives us the degree of x, which, according to my calculations is about 40.29