solve cos^2(x) – 5cos(x) – 1 = 0

Question

approximate solutions to the equationin the interval [0,2π].

Answer 1

The trigonometric equation is cos ²( x ) - 5 cos ( x )  - 1 = 0  and interval [ 0 , 2 π ]

Compare to the quadratic equation form a ² x + b x  +  c.

a = 1 , b = - 5 and c = - 1.

Substituting the a, b and c values

 

The function cos x  has period of 2π , first find all solutions in the interval [ 0 , 2 π ] .

The function cos x  is positive in first and fourth quadrant.

In first quadrant : 

Finally, add multiples of 2 π to each of these solutions to get the general form

In fourth  quadrant :

Finally, add multiples of 2 π to each of these solutions to get the general form

The solutions are