general solution cos (4x-60)=sin2x [-90:90]

Wednesday, August 4, 2010

Solve $\displaystyle{\cos{{\left({4}{x}-{{60}}^{\circ}\right)}}}={\sin{{2}}}{x}$ :

$\displaystyle{\cos{{\left({4}{x}-{{60}}^{\circ}\right)}}}={\sin{{2}}}{x}$

$\displaystyle{\sin{{\left({4}{x}+{{30}}^{\circ}\right)}}}={\sin{{2}}}{x}$

Take the inverse sine of both sides:

1. $\displaystyle{4}{x}+{30}={180}-{2}{x}+{360}{n}$ where n is an integer.

6x=150+360n

x=25+60n

2. 4x+30=2x+360n

2x=-30+360n

x=-15+180n

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The general solution is x=25+60n for n an integer or x=-15+180n for n an integer.

On the interval [-90,90] the solutions are x=-35,-15,25,85

Here is the graph: the left side in black and the right side in red:

convert the scale from radian to degrees.

Ninenotes