General solutions for Cos... (1 Viewer)

[ashokkumar]

Can someone explain how to solve the general solution for cos(x) = -1/2

I don't know how to derive it without using the formula.

 

[Trebla]

Can someone explain how to solve the general solution for cos(x) = -1/2

I don't know how to derive it without using the formula.

cos x = -1/2
First two solutions within single revolution i.e. 0 < x < 2π are:
x = 2π/3, 4π/3
=> x = 2π/3, 2π - 2π/3
Rotating 2π in positive direction:
x = 2π + 2π/3, 4π - 2π/3
Rotating another 2π in positive direction:
x = 4π + 2π/3, 6π - 2π/3

Hence some of the solutions are:
x = 2π/3, 2π + 2π/3, 4π + 2π/3, 6π + 2π/3, etc

By inspection, solutions are:

2nπ + 2π/3

for some integer n

 

[ashokkumar]

How did you know to do 2pi - 2pi/3 in the first revolution.... Because cosx = -1/2 should only be in the 2nd and 3rd quadrant shouldn't it?

 

[fixing]

How did you know to do 2pi - 2pi/3 in the first revolution.... Because cosx = -1/2 should only be in the 2nd and 3rd quadrant shouldn't it?

2pi - 2pi/3 is in the third quadrant dumbass.

 

[ashokkumar]

what i meant was... in the third quadrant its pi + theta, and in the fourth 2pi - theta.

 

[Trebla]

This is true only if theta is acute. In this case, theta is obtuse so theta and 2pi - theta occur in different quadrants.