Cos(2theta)=-1/2
There are 6 solutions, how do i solve this?
cos (2Ø) = -1/2
we know cos 60° = + 1/2
and we know that 2Ø, our angle, must be in quadrants II or III by the CAST rule
so 2Ø = 180-60 = 120° or 2Ø = 180+60 = 240°
Ø = 60° or Ø = 120°
but the period of cos 2Ø is 180°
so adding 180° to our answers will produce more answers.
Ø = 60+180 = 240° or
Ø = 120+180 = 300°
so Ø= 60,120,240,300°
test by doubling any of these answers and taking the cosine on your calculator.
your statement that there are 6 solutions is not really correct, there is an infinite number of solutions, we can keep adding/subtracting 180 to any of our new answers to obtain more answers.
There are 6 solutions in the usual domain between 0° and 360°
To solve the equation cos(2θ) = -1/2, you need to find the values of θ that satisfy the equation. There are a total of six solutions you are looking for.
To solve this equation, you can use the inverse trigonometric function, specifically the arccosine (cos^-1), along with the properties of trigonometric identities.
1. First, take the inverse cosine (arccos) of both sides of the equation to isolate θ:
arccos(cos(2θ)) = arccos(-1/2)
2. The inverse trigonometric function (arccos) will give you the primary solution for 2θ:
2θ = arccos(-1/2)
3. To find the general solution for θ, you need to consider the periodicity of the cosine function. Since the cosine function repeats every 2π radians, you can add 2πn (where n is an integer) to the primary solution to obtain the general solution:
2θ = arccos(-1/2) + 2πn
4. Now you can divide the equation by 2 to solve for θ:
θ = (arccos(-1/2) + 2πn) / 2
5. Simplify the equation to get the general solution for θ:
θ = (arccos(-1/2)/2) + πn
6. Finally, substitute different integer values for n (0, 1, 2, etc.) to find the six unique solutions for θ.
Note: The combination of arccos(-1/2)/2 represents the constant term in the general solution, and πn accounts for the multiple solutions.
By following these steps, you can determine the values of the six solutions for θ.