Which Law or Theorem Would You Use? Law of Cosines Why? You are given the lengths of two sides and the measure of the included angle. Remaining Side and Angles: c≈ 19.3, m∠A≈ 34.3^(∘), m∠B≈ 80.7^(∘)
Practice makes perfect
Let's start by sketching △ ABC and labeling sides a, b, and angle ∠C. Remember that m∠C= 65^(∘), a= 12, and b= 21.
We will solve △ ABC. This means we will find the values of c, m∠A, and m∠B. First, let's find the length of the remaining side and then move on to the measures of the remaining angles.
Remaining Side
Note that we are given the lengths of two sides of â–³ ABC and the included angle.
To find the length c of the third side, we will use the Law of Cosines. Let's recall it.
Law of Cosines
If â–³ ABC has sides of length a, b, and c, then the following are true. a^2=b^2+c^2-2bccosA b^2=a^2+c^2-2accosB c^2=a^2+b^2-2abcosC
In our case we know that a= 12, b= 21, and m∠C= 65^(∘), so we will use the third equation to find c.
The value of c is about 19.3 units. Let's update our diagram.
Remaining Angles
Next we will find the measures of the two remaining angles. Let's start with m∠A. Since now we know the lengths of all three sides of △ ABC and the measure of one angle, we can use the Law of Sines.
Law of Sines
For any triangle the ratio between the sine value of an angle and the length of its opposite side is constant.
Let's apply the Law of Sines to â–³ ABC.
sinA/12=sinB/21=sin 65^(∘)/19.3
Now we will write an equation involving the sine of ∠A.
sinA/12=sin65^(∘)/19.3 ⇔ sinA=12sin65^(∘)/19.3
To find the measure of ∠A, let's use the inverse sine ratio.
sinA=12sin65^(∘)/19.3
⇕
m∠A=sin^(- 1)(12sin65^(∘)/19.3)
We will use a calculator to approximate m∠A to the nearest tenth of a degree.
m∠A=sin^(- 1)(12sin65^(∘)/19.3)≈ 34.3^(∘)
To find the measure of ∠B, we could solve an equation involving the sine of ∠B or we can use the Triangle Sum Theorem.
m∠B≈ 180^(∘)-34.3^(∘)- 65^(∘)=80.7^(∘)
The value of m∠B is about 80.7^(∘).
