What values for θ (0≤θ≤2π) satisfy the equation 2cos θ+1=-cos θ?
To solve the equation 2cos θ + 1 = -cos θ, we can start by simplifying it.
First, let's move the -cos θ to the left side of the equation by adding cos θ to both sides:
2cos θ + cos θ + 1 = 0
Next, combine like terms on the left side:
3cos θ + 1 = 0
Now, let's isolate cos θ by moving the constant term to the right side:
3cos θ = -1
Finally, divide both sides of the equation by 3 to solve for cos θ:
cos θ = -1/3
To find the values for θ (0≤θ≤2π) that satisfy this equation, we need to find the angles where the cosine function is equal to -1/3.
In the unit circle, cos θ represents the x-coordinate of the point on the circle. So, we need to find the angles where the x-coordinate is equal to -1/3.
Depending on the specific instructions of the problem, you may be asked to provide the answer in degrees or radians. Here, I will explain how to find the solutions in both degrees and radians.
Degrees:
To find the values for θ in degrees, we can use the inverse cosine function (also known as arccos) to find the angle. The inverse cosine function will give us the principal value of the angle between 0° and 180°.
Use a calculator or a table of trigonometric values to find the principal value of the angle where cos θ = -1/3. In this case:
θ ≈ 109.471°
Since θ needs to be between 0° and 360°, we need to find all possible angles that satisfy the equation. We can do this by adding or subtracting multiples of 360° to the principal value.
The possible solutions in degrees are:
θ ≈ 109.471° + k * 360°
where k is an integer.
Radians:
To find the values for θ in radians, we can also use the inverse cosine function (arccos). The inverse cosine function will give us the principal value of the angle between 0 and π (180°).
Use a calculator or a table of trigonometric values to find the principal value of the angle where cos θ = -1/3. In this case:
θ ≈ 1.9106 radians
To find all possible angles between 0 and 2π (360°), we can add or subtract multiples of 2π:
The possible solutions in radians are:
θ ≈ 1.9106 + k * 2π
where k is an integer.
Thus, the values for θ (0≤θ≤2π) that satisfy the equation 2cos θ+1=-cos θ are approximately:
θ ≈ 109.471° + k * 360° (in degrees)
θ ≈ 1.9106 + k * 2π (in radians)