Solve this multiple-angle equation.
cos x/2 = -(sqrt3)/2
well, you know that cos π/6 = √3/2
So, with the negative value, you have
x/2 = 5π/6 or 7π/6
So the solutions are also 5π/6 + 2npi and 7π/6 2npi, right?
nope. cos(x/2) has period 4π
also, note that x/2 = 5π/6. So, x = 5π/3.
Watch the details, guy.
To solve the multiple-angle equation cos(x/2) = -(sqrt(3)/2), we can use the trigonometric identity for the cosine of half angles.
The identity for cos(x/2) states that cos(x/2) = ± sqrt((1 + cos(x))/2).
In this case, we know that cos(x/2) = -(sqrt(3)/2). So let's substitute the given value into the identity:
-(sqrt(3)/2) = ± sqrt((1 + cos(x))/2).
To proceed, we will square both sides of the equation to eliminate the square root:
((-sqrt(3)/2))^2 = (sqrt((1 + cos(x))/2))^2
3/4 = (1 + cos(x))/2
Next, we will simplify the equation by multiplying both sides by 2:
2 * (3/4) = 2 * (1 + cos(x))/2
3/2 = 1 + cos(x)
Now, isolate the cos(x) term:
cos(x) = 3/2 - 1
cos(x) = 1/2
To find the solutions for x, we need to consider the values of the cosine function within the given domain. In this case, cos(x) = 1/2 when x is equal to π/3 or x is equal to 5π/3.
Therefore, the solutions for the given multiple-angle equation cos(x/2) = -(sqrt(3)/2) are x = π/3 + 2πn and x = 5π/3 + 2πn, where n is an integer.