Find all solutions to 2 cos(theta) = - 1 on the interval 0 less than or equal to theta less than 2 pi
Give your answers as exact values, as a list separated by commas.
cosine is negative in II or III
and I know cos 60° = cosØ/3 = 1/2
so for your cosØ = -1/2
Ø = 180-60 =120° or Ø = 180+60 = 240°
in radians
Ø = 2π/3 or Ø = 4π/3
To find all the solutions to the equation 2cos(theta) = -1 on the interval 0 ≤ theta < 2π, we can follow these steps:
Step 1: Isolate the cosine term.
Divide both sides of the equation by 2:
cos(theta) = -1/2.
Step 2: Find the reference angle.
The reference angle is the positive acute angle formed between the terminal arm and the x-axis in the unit circle when cos(theta) = -1/2. In this case, the reference angle is π/3.
Step 3: Determine the possible values of theta.
To find the values of theta on the interval 0 ≤ theta < 2π, we can consider the location of the reference angle (π/3) in the unit circle:
- In the first quadrant, the cosine is positive. Therefore, the angle θ that satisfies cos(theta) = -1/2 will be π - π/3 = 2π/3.
- In the second quadrant, the cosine is negative. Therefore, the angle θ that satisfies cos(theta) = -1/2 will be π + π/3 = 4π/3.
Step 4: List all the solutions.
Since the interval is 0 ≤ theta < 2π, the solutions are 2π/3 and 4π/3.
So, the exact values of theta that satisfy the equation 2cos(theta) = -1 on the interval 0 ≤ theta < 2π are:
θ = 2π/3, 4π/3.