Question

find the length of side a.
b = 4
c = 10
a = 29°
a = ?
round to the nearest tenth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).
Substitute \(b = 4\), \(c = 10\), and \(A=29^{\circ}\) into the formula:
\(a^{2}=4^{2}+10^{2}-2\times4\times10\times\cos(29^{\circ})\).

Step2: Calculate each term

First, \(4^{2}=16\), \(10^{2}=100\).
\(\cos(29^{\circ})\approx0.8746\), then \(2\times4\times10\times\cos(29^{\circ}) = 80\times0.8746=69.968\).
So \(a^{2}=16 + 100-69.968\).

Step3: Simplify the right - hand side

\(a^{2}=116 - 69.968=46.032\).

Step4: Solve for \(a\)

Take the square root of both sides: \(a=\sqrt{46.032}\approx6.8\).

Answer:

\(6.8\)