Which Law or Theorem Would You Use? Law of Sines Why? You are given the lengths of two sides and the measure of a nonincluded angle. Remaining Angles and Side: m∠A≈ 111.2^(∘), m∠B≈ 28.8^(∘), a≈ 52.2
Practice makes perfect
Let's start by sketching △ ABC and labeling sides b, c, and angle ∠C. Remember that m∠C= 40^(∘), b= 27, and c= 36.
We will solve △ ABC. This means we will find the values of a, m∠A, and m∠B. First, let's find the measures of the remaining angles and then move on to the length of the remaining side.
Remaining Angles
Since we are given the lengths of two sides and the measure of a nonincluded angle, we can use the Law of Sines to find the measure of the other nonincluded angle, ∠B. Let's recall the law.
Law of Sines
For any triangle the ratio between the length of the side and the sine value of its opposite angle is constant.
Now we will apply the Law of Sines to the given triangle.
a/sinA=27/sinB=36/sin 40^(∘)
Let's write an equation involving the sine of ∠B.
27/sinB=36/sin40^(∘)
We will rewrite the obtained equation so that sinB is isolated on the left-hand side.
To find the measure of ∠B, let's use the inverse sine ratio.
sinB=3sin40^(∘)/4
⇕
m∠B=sin^(- 1)(3sin40^(∘)/4)
We will use a calculator to approximate m∠B to the nearest tenth of a degree.
m∠B=sin^(- 1)(3sin40^(∘)/4)≈28.8^(∘)
Finally to find the measure of ∠A, we will use the Triangle Sum Theorem.
m∠A≈ 180^(∘)-28.8^(∘)- 40^(∘)=111.2^(∘)
The measure of ∠A is about 111.2^(∘). Let's update our diagram.
Remaining Side
Note that we know the measures of all three angles of â–³ ABC and the lengths of two of its sides. To find the length a of the remaining side, we will use the Law of Sines once more.
a/sin111.2^(∘)≈36/sin 40^(∘) [0.7em]
⇕ [0.7em]
a≈36sin11.2^(∘)/sin40^(∘)
Let's use a calculator to approximate the value of a.
a≈36sin111.2^(∘)/sin40^(∘)≈ 52.2
The length a of the remaining side is about 52.2.
