Evaluate the measures of the player's angles using the Law of Cosines.
Which player has a greater chance to make a shot? Alyssa The measures of the player's angles: Alyssa 28.2^(∘), Nari 13.5^(∘)
Practice makes perfect
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.
12^2= 20^2+ 25^2-2( 20)( 25)cos 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.
cos A=0.881 ⇓
A=cos^(-1)0.881≈28.2^(∘)
Alyssa's angle is approximately 28.2^(∘).
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.
12^2= 45^2+ 38^2-2( 45)( 38)cos N
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.
cos N=665/684 ⇓
N=cos^(-1)665/684≈13.5^(∘)
Nari's angle is approximately 13.5^(∘).
Comparison
We found that Alyssa has an angle measure of about 28.2^(∘) and Nari's angle is about 13.5^(∘). 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.
