Question

Find the angle θ (in radians) such that sin(θ) = −(2/7)
,cos(θ) < 0, and 2π ≤ θ ≤ 4π

Answer 1

To find the angle θ in radians with sin(θ) = −(2/7) and cos(θ) < 0 in the interval from 2π to 4π, we find the reference angle in the first quadrant, add π to shift it to the third quadrant, and finally add 2π to make sure it falls within the specified interval.

Step-by-step explanation:

The question asks us to find the angle θ (in radians) given that sin(θ) = −(2/7), cos(θ) < 0, and 2π ≤ θ ≤ 4π. Since sin(θ) is negative and cos(θ) is negative as well, the angle θ must be in the third quadrant, where both sine and cosine are negative. However, because the angle must also be between 2π and 4π, after finding the reference angle in the first quadrant, we add 2π to find the angle in the desired interval.

To find θ, we first find the reference angle α such that sin(α) = 2/7. This reference angle α is in the first quadrant. Since no angle is provided, we'll use a calculator to find α = sin⁻¹(2/7). Now, to find the angle in the third quadrant, we take π + α. This gives us the angle in the third quadrant during the first circulation starting from 0. As we need the angle between 2π and 4π, we add 2π to this result, thereby obtaining θ = π + α + 2π. Finally, we convert θ into radians, if necessary, since our reference angle α was found using a calculator which likely provided an answer in radians.

Answer 2

To find the angle θ such that sin(θ) = -(2/7) and cos(θ) < 0, we can use the inverse sine function and convert the degree measure to radians.

Step-by-step explanation:

To find the angle θ such that sin(θ) = -(2/7), we can use the inverse sine function, also known as arcsin or sin-1. Since cos(θ) < 0, we know that θ lies in the second or third quadrant. The given range of 2π ≤ θ ≤ 4π tells us that θ is between 360 and 720 degrees. To find the angle in radians, we can convert the degree measure to radians by multiplying by π/180.

Step 1: Use arcsin to find θ in degrees:

θ in degrees = arcsin(-(2/7)) = -43.29 degrees.

Step 2: Convert to radians:

θ in radians = -43.29 * (π/180) = -0.756 radians.