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 the values into the formula:
\( a^{2}=2^{2}+4^{2}-2\times2\times4\times\cos(78^{\circ}) \).
First, calculate the squares: \( 2^{2} = 4 \), \( 4^{2}=16 \).
And \( \cos(78^{\circ})\approx0.2079 \).
Then calculate \( 2\times2\times4\times0.2079 = 3.3264 \).
So \( a^{2}=4 + 16-3.3264 \).

Step2: Simplify the right - hand side

\( a^{2}=20 - 3.3264=16.6736 \).

Step3: Solve for \( a \)

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

Answer:

\( 4.1 \)