Question

Given ΔABC with m∠B = 74°, a = 38, and c = 41, what is the measure of A? m∠A = 55.9° m∠A = 51.9° m∠A = 50.1° m∠A = 54.1°

Answer 1

We will use the following trigonometric laws, sine law and cosine law.

Cosine law is :

`c^2=a^2+b^2-2ab\cos C`

Sine law is :

`(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)`

From the problem, we have B = 74 degrees, a = 38 and c = 41

We need first to find the value of b using cosine law, note that the missing value is b, so we will rewrite the cosine law as b in terms of a and c :

`b^2=a^2+c^2-2ac\cos B`

Substitute the given values and solve for b :

`\begin{gathered} b^2=38^2+41^2-2(38)(41)\cos 74 \\ b^2=3125-3116\cos 74 \\ b=\sqrt[]{3125-3116\cos 74} \\ b=47.60 \end{gathered}`

Next is to use the sine law with B = 74 degrees, b = 47.60 and a = 38.

`\begin{gathered} (b)/(\sin B)=(a)/(\sin A) \\ (47.60)/(\sin 74)=(38)/(\sin A) \\ \sin A=(38\sin 74)/(47.60) \\ \sin A=0.7674 \\ \arcsin (\sin A)=\arcsin (0.7674) \\ A=50.12 \end{gathered}`

The answer is C. m∠A = 50.1 degrees