Which Law or Theorem Would You Use? Law of Cosines Why? You are given the lengths of all three sides. Remaining Angles: m∠ A≈ 93.7^(∘), m∠ B=33.9^(∘), m∠ C≈ 52.4^(∘)
Practice makes perfect
Let's start by sketching △ ABC and labeling the sides a, b, and c. Remember that a= 34, b= 19, and c= 27.
We will solve △ ABC. This means we will find the values of all three angles of the triangle. Since we are given the lengths of all three sides of △ ABC, we will use the Law of Cosines. Let's recall it.
Law of Cosines
If △ 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
First we will find the measure of ∠ A. To do it, we will use the first equation which includes the cosine of ∠ A.
a^2=b^2+c^2-2bccosA
Let's substitute the values of a, b, and c into the above equation and simplify it so that cosA is isolated on the left-hand side.
To find the measure of ∠ A, let's use the inverse cosine ratio.
cosA=-66/1026 ⇔ m∠ A=cos^(- 1)(-66/1026)
We will use a calculator to approximate m∠ A to the nearest tenth of a degree.
m∠ A=cos^(- 1)(-66/1026)≈ 93.7^(∘)
Similarly, let's find the value of m∠ B. This time we will use the second equation from the Law of Cosines.
The measure of ∠ B is about 33.9^(∘). To find the measure of the remaining angle, let's use the Triangle Sum Theorem.
m∠ C≈ 180^(∘)- 93.7^(∘)- 33.9^(∘)=52.4^(∘)
The measure of ∠ C is about 52.4^(∘).
