Question

find the length of side a.
b = 2
c = 4
a = 78°
a = ?

Explanation:

Step1: Apply Law of Cosines

The Law of Cosines for side $a$ is $a^2 = b^2 + c^2 - 2bc\cos(A)$.
Substitute given values: $b=2$, $c=4$, $A=78^\circ$
$a^2 = 2^2 + 4^2 - 2\times2\times4\times\cos(78^\circ)$

Step2: Calculate each term

$2^2=4$, $4^2=16$, $2\times2\times4=16$, $\cos(78^\circ)\approx0.2079$
$a^2 = 4 + 16 - 16\times0.2079$

Step3: Simplify the expression

$4+16=20$, $16\times0.2079\approx3.3264$
$a^2 \approx 20 - 3.3264 = 16.6736$

Step4: Take square root

$a \approx \sqrt{16.6736}$

Answer:

$\approx 4.08$ (rounded to two decimal places, or 4 if rounded to nearest whole number)