find all solutions of the equation in the interval 0, 2π). cotθ = -5/4 if there is more than one solution…

find all solutions of the equation in the interval 0, 2π). cotθ = -5/4 if there is more than one solution, separate them with commas. do not round any intermediate computations. give your answer(s) in radians, and round your answer(s) to the nearest hundredth.

find all solutions of the equation in the interval 0, 2π). cotθ = -5/4 if there is more than one solution, separate them with commas. do not round any intermediate computations. give your answer(s) in radians, and round your answer(s) to the nearest hundredth.

Answer

Explanation:

Step1: Recall the relationship between cotangent and tangent

Since (\cot\theta=\frac{1}{\tan\theta}), the equation (\cot\theta =-\frac{5}{4}) can be rewritten as (\tan\theta=-\frac{4}{5}).

Step2: Find the reference angle

The reference angle (\theta_{ref}=\arctan\left(\frac{4}{5}\right)). Using a calculator, (\theta_{ref}\approx 0.67) radians.

Step3: Determine the solutions in the interval ([0, 2\pi))

The tangent function (y = \tan\theta) is negative in the second and fourth quadrants.

  • For the second - quadrant solution: (\theta=\pi-\theta_{ref}). Substituting (\theta_{ref}\approx0.67), we get (\theta=\pi - 0.67\approx2.47) radians.
  • For the fourth - quadrant solution: (\theta = 2\pi-\theta_{ref}). Substituting (\theta_{ref}\approx0.67), we get (\theta=2\pi - 0.67\approx5.61) radians.

Answer:

(2.47,5.61)