Question

in $\triangle abc$, $ab = 30$, $ac = 23$, and $bc = 19$. what is $m\angle a$?

$m\angle a = \square^{\circ}$
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Identify the Law to Use

We have a triangle with three sides given, and we need to find the measure of an angle. The Law of Cosines is appropriate here. The Law of Cosines formula for angle \( A \) in \( \triangle ABC \) is \( BC^{2}=AB^{2}+AC^{2}-2\cdot AB\cdot AC\cdot\cos(A) \).

Given \( AB = 30 \), \( AC = 23 \), and \( BC = 19 \). Substitute these values into the formula:
\( 19^{2}=30^{2}+23^{2}-2\cdot30\cdot23\cdot\cos(A) \)

Step2: Simplify the Equation

First, calculate the squares:
\( 361 = 900 + 529 - 1380\cdot\cos(A) \)
Then, add \( 900 \) and \( 529 \):
\( 361 = 1429 - 1380\cdot\cos(A) \)

Step3: Solve for \( \cos(A) \)

Subtract \( 1429 \) from both sides:
\( 361 - 1429=- 1380\cdot\cos(A) \)
\( - 1068=- 1380\cdot\cos(A) \)
Divide both sides by \( - 1380 \):
\( \cos(A)=\frac{-1068}{-1380}=\frac{1068}{1380} \)
Simplify the fraction: \( \frac{1068\div12}{1380\div12}=\frac{89}{115}\approx0.7739 \)

Step4: Find the Angle \( A \)

Take the inverse cosine (arccos) of \( 0.7739 \):
\( A=\arccos(0.7739) \)
Using a calculator, \( A\approx39.3^{\circ} \) (rounded to the nearest tenth)

Answer:

\( 39.3 \)