Solve the equation \(5\cos (6x - 20)^\circ + 3 = 7.25\), for \(0 \le x \le 180\).

\(5\cos (6x - 20)^\circ + 3 = 7.25\)

\(5\cos (6x - 20) = 4.25\)

\(\cos (6x - 20) = 0.85\)

\(6x - 20 = 31.8^\circ\)

\(6x = 51.8^\circ\)

\(x = 8.6^\circ\)

\(6x - 20 = 360^\circ - 31.8^\circ\)

\(6x - 20 = 328.2^\circ\)

\(6x = 348.2^\circ\)

\(x = 58.0^\circ\)

Since \(0 \le x \le 180\), then we need to find out the other results by adding the period to these solutions.

\(Period = 360^\circ \div 6 = 60^\circ\)

3rd solution: \(8.6 + 60 = 68.6^\circ\)

4th solution: \(58.0 + 60 = 118^\circ\)

5th solution: \(68.6 + 60 = 128.6^\circ\)

6th solution: \(118 + 60 = 178^\circ\)

7th solution: \(128.6 + 60 = 188.6^\circ\). This is not a solution since \(0 \le x \le 180\).

Therefore \(x^\circ = 8.6^\circ ,\,58^\circ ,\,68.6^\circ ,\,118^\circ,\,128.6^\circ,\,178^\circ,\,188.6^\circ\)