Answer:
The solutions are 7π/6 and 11π/6
Step-by-step explanation:
Given the equation sec(x)cot(x)= -2 on the interval [0,2π]. In order to get the value of x, the following steps must be followed.
sec(x)cot(x)= -2
From trigonometry identity, sec(x)= 1/cos(x) and cot(x)= 1/tan(x) = cos(x)/sin(x)
subsituting this expressions in the given equation we have;
1/cos(x)*cos(x)/sin(x) = -2
1/sin(x) = -2
sin(x) = -1/2
[tex]x = sin^{-1}-1/2\\ x = -30^{0}[/tex]
Since sin is negative in the 3rd and 4th quadrant,
In the 3rd quadrant, x = 180+30 = 210°
In the 4th quadrant, x = 360-30 = 330°
Converting the values to radian;
since 180° = πrad
210° = 210π/180 rad
210° = 7π/6 rad
Similarly, 330° = 330π/180 rad
330° = 11π/6 rad
The solutions are 7π/6 and 11π/6
