Exercise 44 Page 595

Let's begin with recalling the Law of Cosines. If $△ABC$ has lengths of $a,b,$ and $c$ and angle measures of $A,B,$ and $C,$ then we can write equations that relate the side lengths of this triangle and the cosine of one of its angles.

In our exercise we are given that Alyssa and Nari are playing field hockey and asked to determine which of them has a greater chance to make a shot. To do this, we will compare the measures of the players' angles. We will start with evaluating the measure of Alyssa's angle.

Alyssa

We know that Alyssa is standing $20$ feet from the goal post and $25$ feet from the opposite post. We are also given that the goal is $12$ feet wide. Let $A$ represents the measure of Alyssa's angle.

Using the Law of Cosines, we can write an equation for $A.$ Let's solve the equation. We will start with isolating $\cos A.$ Next let's use the inverse cosine to find the value of $A.$ Alyssa's angle is approximately $28.2^{\circ }.$

Nari

We are given that Nari is standing $45$ feet from the goal post and $38$ feet from the opposite post. The goal is $12$ feet wide. Let $N$ represents the measure of Nari's angle.

Again, we can write an equation for $N$ using the Law of Cosines. Let's solve the equation. We will start with isolating $\cos N.$ Next let's use the inverse cosine to find the value of $N.$ Nari's angle is approximately $13.5^{\circ }.$

Comparison

We found that Alyssa has an angle measure of about $28.2^{\circ }$ and Nari's angle is about $13.5^{\circ }.$ Since Alyssa's angle is greater than than Nari's angle, Alyssa has a wider target and so she has a greater chance to make the shot.