Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$(\cot \theta+1)(\sin \theta-1)=0$[/tex]
Select the correct choice below and fill in any answer boxes in your choice.
A. The solution set is { }. (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. Type your answer in radians. expression. Use a comma to separate answers as needed.)
B. There is no solution on this interval.
Answer (2)
Set each factor to zero: cot θ + 1 = 0 or sin θ − 1 = 0 .
Solve cot θ = − 1 , which gives θ = 4 3 π and θ = 4 7 π .
Solve sin θ = 1 , which gives θ = 2 π .
The solution set is 2 π , 4 3 π , 4 7 π .
Explanation
Understanding the Problem We are given the equation ( cot θ + 1 ) ( sin θ − 1 ) = 0 and asked to solve for θ in the interval 0 ≤ θ < 2 π . This means we need to find all values of θ within this interval that make the equation true.
Setting Factors to Zero To solve the equation, we set each factor equal to zero: cot θ + 1 = 0 or sin θ − 1 = 0
Solving cot(θ) = -1 First, let's solve cot θ + 1 = 0 . This is equivalent to cot θ = − 1 . Recall that cot θ = s i n θ c o s θ , so we are looking for angles where cos θ = − sin θ . This occurs in the second and fourth quadrants. The reference angle is 4 π , so the solutions in the interval 0 ≤ θ < 2 π are θ = π − 4 π = 4 3 π and θ = 2 π − 4 π = 4 7 π
Solving sin(θ) = 1 Next, let's solve sin θ − 1 = 0 . This is equivalent to sin θ = 1 . The only angle in the interval 0 ≤ θ < 2 π for which sin θ = 1 is θ = 2 π
Combining the Solutions Therefore, the solutions to the equation ( cot θ + 1 ) ( sin θ − 1 ) = 0 in the interval 0 ≤ θ < 2 π are θ = 4 3 π , 4 7 π , 2 π .
Final Answer The solution set is { 2 π , 4 3 π , 4 7 π } .
Examples
Imagine you're designing a robotic arm that needs to reach specific angles to perform tasks. Solving trigonometric equations like this helps you determine the exact angles the arm needs to move to, ensuring it interacts correctly with its environment. For example, if the arm's movement is described by trigonometric functions, finding the solutions to these equations tells you the precise positions the arm can achieve. This is crucial for automation in manufacturing, surgery, and exploration, where precision is key.
The solution set for the equation ( cot θ + 1 ) ( sin θ − 1 ) = 0 in the interval 0 ≤ θ < 2 π is { 2 π , 4 3 π , 4 7 π }.
;
