Solve the equation cos (3θ +120°) = √(3 /2 for 0° ≤ θ≤ 180°
cos(x) = √3/2 if x = 30° or 330°
So,
3θ+120 = 30, 330, 390, 690, ...
θ = -30, 70, 90, 190, ...
So, pick the values in your domain.
To solve the equation cos(3θ + 120°) = √(3/2) for 0° ≤ θ ≤ 180°, we can follow these steps:
Step 1: Simplify the right side of the equation.
√(3/2) = cos(π/6)
Step 2: Equate the angles.
3θ + 120° = π/6
Step 3: Solve for θ.
3θ = π/6 - 120°
3θ = π/6 - (720°/6)
3θ = π/6 - 120°/6
3θ = π/ 6 - 20°
Step 4: Convert the angles to a common unit (radians or degrees). Let's solve it in radians first.
3θ = π/6 - 20° * π/180
3θ = π/6 - (20π/180)
3θ = π/6 - (π/9)
3θ = (π - 2π)/18
3θ = -π/18
Step 5: Divide both sides of the equation by 3.
θ = -π/54
Step 6: Convert the solution back to degrees.
θ = -π/54 * (180/π)
θ ≈ -3.33°
Therefore, the solution to the equation cos(3θ + 120°) = √(3/2) for 0° ≤ θ ≤ 180° is θ ≈ -3.33°.
To solve the equation cos(3θ + 120°) = √(3/2) for θ between 0° and 180°, we can utilize trigonometric identities and properties to simplify and solve the equation.
1. Start by taking the inverse cosine (arccos) of both sides of the equation:
arccos(cos(3θ + 120°)) = arccos(√(3/2))
2. Apply the property of inverse cosine and cosine to simplify the equation:
3θ + 120° = arccos(√(3/2))
3. Since we want to solve for θ, isolate θ by subtracting 120° from both sides:
3θ = arccos(√(3/2)) - 120°
4. Divide both sides of the equation by 3 to solve for θ:
θ = (arccos(√(3/2)) - 120°) / 3
5. Plug in the value of √(3/2) into a calculator to find its numerical value:
√(3/2) ≈ 0.866
6. Substitute this value into the equation and solve for θ:
θ = (arccos(0.866) - 120°) / 3
7. Use a calculator to find the numerical value of arccos(0.866):
arccos(0.866) ≈ 30°
8. Substitute the value into the equation and calculate θ:
θ = (30° - 120°) / 3
θ = -90° / 3
θ = -30°
Therefore, the solution to the equation cos(3θ + 120°) = √(3/2) for θ between 0° and 180° is θ = -30°.