Trigonometric Equations involving compound angles (extension)
Example
Solve the equation \(5\cos (6x - 20)^\circ + 3 = 7.25\), for \(0 \le x \le 180\).
Solution
\(5\cos (6x - 20)^\circ + 3 = 7.25\)
\(5\cos (6x - 20) = 4.25\)
\(\cos (6x - 20) = 0.85\)
Since cos is positive, we are in the 1st and 4th quadrants.
First quadrant
\(6x - 20 = 31.8^\circ\)
\(6x = 51.8^\circ\)
\(x = 8.6^\circ\)
Fourth quadrant
\(6x - 20 = 360^\circ - 31.8^\circ\)
\(6x - 20 = 328.2^\circ\)
\(6x = 348.2^\circ\)
\(x = 58.0^\circ\)
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 \(x^\circ = 8.6^\circ ,\,58^\circ ,\,68.6^\circ ,\,118^\circ,\,128.6^\circ,\,178^\circ,\,188.6^\circ\)