What is the measure of ∠S to the nearest whole degree?
Use: $a^2=b^2+c^2-2 b c \cos (A)$
To determine the measure of \(\angle S\) in a triangle, we use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles:
\[s^2 = r^2 + t^2 - 2rt \cos(S)\]
where \(s\) is the side opposite to \(\angle S\), and \(r\) and \(t\) are the other two sides of the triangle. To solve for \(\cos(S)\), we rearrange the formula:
\[\cos(S) = \frac{r^2 + t^2 - s^2}{2rt}\]
Given a triangle with side lengths \(r=7\), \(t=8\), and \(s=4\), we can substitute these values into the formula to find the cosine of the angle.
Using the provided side lengths \(r=7\), \(t=8\), and \(s=4\), we substitute these into the expression:
\[\cos(S) = \frac{7^2 + 8^2 - 4^2}{2(7)(8)}\]
Calculate the squares:
\[\cos(S) = \frac{49 + 64 - 16}{112}\]
Perform the addition and subtraction in the numerator:
\[\cos(S) = \frac{97}{112}\]
Now, calculate the value of the fraction:
\[\cos(S) \approx 0.86607\]
To find the measure of \(\angle S\), we apply the inverse cosine function (\(\arccos\)):
\[S = \arccos(0.86607)\]
Using the calculator, the result is:
\[S \approx 29.99^\circ\]
Rounding this to the nearest whole degree gives:
\[S \approx 30^\circ\]
Thus, the measure of \(\angle S\) is \(30^\circ\).
The Law of Cosines is essential in fields like engineering and navigation. For example, if two surveyors are measuring distances from a single point to two distant landmarks (forming two sides of a triangle) and they measure the angle between the lines of sight, they can use the Law of Cosines to calculate the distance between the two landmarks without having to travel that distance directly. This is a practical way to map terrain, survey plots of land, or determine the distance between two ships at sea using only angle measurements and known baselines. If you have sides \(a=10\) km and \(b=12\) km with an included angle of \(60^\circ\), you can find the third side \(c\) using \(c^2 = 10^2 + 12^2 - 2(10)(12)\cos(60^\circ) = 100 + 144 - 120 = 124\), so \(c = \sqrt{124} \approx 11.14\) km.