Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$(\tan \theta-1)(\sec \theta-1)=0$[/tex]
Select the correct choice below and fill in any answer boxes.
A. The solution set is $\square$ (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. Use a comma to separate answers as needed.)
Answer (1)
Set each factor to zero: tan θ − 1 = 0 or sec θ − 1 = 0 .
Solve tan θ = 1 for θ in the interval 0 ≤ θ < 2 π , which gives θ = 4 π and θ = 4 5 π .
Solve sec θ = 1 for θ in the interval 0 ≤ θ < 2 π , which gives θ = 0 .
The solution set is 0 , 4 π , 4 5 π .
Explanation
Understanding the Problem We are given the equation ( tan θ − 1 ) ( sec θ − 1 ) = 0 and asked to solve for θ in the interval 0 ≤ θ < 2 π . This means we need to find all angles θ between 0 and 2 π (excluding 2 π ) that make the equation true.
Setting up the Equations The equation is satisfied if either tan θ − 1 = 0 or sec θ − 1 = 0 . Let's solve each of these equations separately.
Solving for tan First, consider tan θ − 1 = 0 , which means tan θ = 1 . The tangent function is equal to 1 at angles where the sine and cosine are equal. In the interval 0 ≤ θ < 2 π , this occurs at θ = 4 π and θ = 4 5 π .
Solving for sec Next, consider sec θ − 1 = 0 , which means sec θ = 1 . Since sec θ = c o s θ 1 , this is equivalent to cos θ = 1 . In the interval 0 ≤ θ < 2 π , this occurs at θ = 0 .
Combining the Solutions Therefore, the solutions to the equation ( tan θ − 1 ) ( sec θ − 1 ) = 0 in the interval 0 ≤ θ < 2 π are θ = 0 , 4 π , 4 5 π .
Final Answer The solution set is { 0 , 4 π , 4 5 π } .
Examples
Imagine you're designing a robotic arm that needs to reach a specific point. The equation you solved helps determine the possible angles the arm can be set at to reach that point. By finding the solutions within a certain range (like 0 to 2 π ), you ensure the arm operates within its physical limits. This is crucial in robotics, engineering, and even animation, where precise movements are required.
