Question

Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0

∘
≤θ<360
∘
, that satisfy the equation below, to the nearest tenth of a degree.
3, cosine, squared, theta, plus, cosine, theta, equals, 0
3cos
2
θ+cosθ=0

Answer 1

To solve the equation 3cos²θ + cosθ = 0 for 0° ≤ θ < 360°, factor it to get cosθ(3cosθ + 1) = 0. The angles satisfying this equation are 90° and 270°, as well as the angles that correspond to cosθ = -1/3, calculated using a cosine inverse function.

Step-by-step explanation:

To find all angles θ such that 0° ≤ θ < 360° satisfying the equation 3cos2θ + cosθ = 0, we can factor the equation as follows:

cosθ(3cosθ + 1) = 0

This gives us two cases:

1. cosθ = 0
2. 3cosθ + 1 = 0 or cosθ = -1/3

In the first case, cosθ = 0 at 90° and 270°. In the second case, cosθ = -1/3, we use a calculator to find the angles, which are not standard angles on the unit circle. Therefore, the angles that satisfy cosθ = -1/3 will be acquired using inverse cosine and will be dependent on the specific calculator or technology used to find the approximate values.

Summarizing, the angles that satisfy the original equation are 90°, 270°, and the angles found by taking cos-1(-1/3) that lie within the given interval.

Answer 2

To find the angles that satisfy the equation 3cos²θ + cosθ = 0, rearrange the equation and solve for cosθ equals 0 and 3cosθ + 1 equals 0. The angles that satisfy the equation are 90°, 109.5°, 250.5°, and 270°.

Step-by-step explanation:

To find the angles that satisfy the equation 3cos2θ + cosθ = 0, we can rewrite it as a quadratic equation:

3cos2θ + cosθ = 0.

Rearranging the terms, we have cosθ(3cosθ + 1) = 0.

This equation is satisfied when cosθ = 0 or

3cosθ + 1 = 0.

For cosθ = 0, we have

θ = 90° and

θ = 270° because those are the angles where the cosine function equals zero.

For 3cosθ + 1 = 0, we can solve for θ by subtracting 1 and dividing by 3 to get:

cosθ = -1/3.

Taking the inverse cosine of both sides, we find that θ is approximately 109.5° and 250.5°.