Question

What are the principle and symmetric solutions to the equation 2cos(x)=-1

Answer 1

Assuming that x is in degrees (you could generalize this to any unit)

For an equation like:

sin(a*x) = K

The principle solutions are the values of x in the range:

0° ≤ x < 360°

And the symmetric ones are the solutions:

xₙ = x + n*360°

where n is an integer.

Ok, first let's solve:

2*cos(x) = -1

isolating the cosine part, we get:

cos(x) = -1/2

Now we can use the inverse cosine function, Acos(x)

If we apply this function to both sides, we get:

Acos (cos(x)) = Acos(-1/2)

x = Acos(-1/2) = 120°

x = 120° is the princpiple solution.

And the symmetric solutions are:

xₙ = 120° + n*360°

So for example:

x₁ = 120° + 1*360° = 480°

x₃ = 120° + 3*360° = 1200°

etc...