Question
Use a calculator to solve each of the following, 0 ≤ 𝜃 ≤ 2𝜋, to the nearest tenth of a radian. b. csc 𝜃 = −2.619
Answers
GPT 3.5
To find the angle 𝜃 where csc 𝜃 = -2.619, we can use the equation csc 𝜃 = 1/sin 𝜃.
Therefore, sin 𝜃 = 1 / (-2.619) = -0.3819.
Using a calculator and the sin^-1 function, we find that sin^-1(-0.3819) ≈ -0.39 (in radians).
Therefore, 𝜃 ≈ -0.39 radians. Since the angle must be between 0 and 2𝜋, we can add 2𝜋 to the angle to find an equivalent angle within the given range.
𝜃 = -0.39 + 2𝜋 ≈ 5.90 radians.
Therefore, the angle where csc 𝜃 = -2.619 is approximately 5.90 radians.
Therefore, sin 𝜃 = 1 / (-2.619) = -0.3819.
Using a calculator and the sin^-1 function, we find that sin^-1(-0.3819) ≈ -0.39 (in radians).
Therefore, 𝜃 ≈ -0.39 radians. Since the angle must be between 0 and 2𝜋, we can add 2𝜋 to the angle to find an equivalent angle within the given range.
𝜃 = -0.39 + 2𝜋 ≈ 5.90 radians.
Therefore, the angle where csc 𝜃 = -2.619 is approximately 5.90 radians.