Question

find the length of side a.
b = 2
c = 4
a = 78°
a = ?
round to the nearest tenth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).
Given \(b = 2\), \(c = 4\), and \(A=78^{\circ}\), substitute these values into the formula:
\(a^{2}=2^{2}+4^{2}-2\times2\times4\times\cos(78^{\circ})\).

Step2: Calculate each term

First, calculate \(2^{2}=4\), \(4^{2}=16\).
Then, calculate \(\cos(78^{\circ})\approx0.2079\).
So, \(2\times2\times4\times\cos(78^{\circ}) = 16\times0.2079 = 3.3264\).
Now, \(a^{2}=4 + 16-3.3264\).
\(a^{2}=20 - 3.3264=16.6736\).

Step3: Find the value of \(a\)

Take the square root of \(a^{2}\): \(a=\sqrt{16.6736}\approx4.1\).

Answer:

\(4.1\)