Which values represent solutions to the equation? [tex]\tan \left(\frac{3 \pi}{4}-2 x\right)=-1[/tex], where [tex]$x \in[0, \pi]$[/tex]
A. [tex]$\left\{0, \pi\right\}$[/tex]
B. [tex]$\left\{0, \frac{\pi}{2}, \pi\right\}$[/tex]
C. [tex]$\left\{0, \pi, 2 \pi\right\}$[/tex]
Answer (2)
Find the general solution for tan ( θ ) = − 1 , which is θ = 4 3 π + nπ .
Substitute θ = 4 3 π − 2 x and solve for x , obtaining x = − 2 nπ .
Determine the valid integer values of n such that x ∈ [ 0 , π ] , which are n = 0 , − 1 , − 2 .
Calculate the corresponding values of x : 0 , 2 π , π . The solution set is { 0 , 2 π , π } .
Explanation
Problem Analysis We are given the equation tan ( 4 3 π − 2 x ) = − 1 , where x ∈ [ 0 , π ] . We need to find the values of x in the interval [ 0 , π ] that satisfy the given equation.
General Solution The general solution for tan ( θ ) = − 1 is θ = 4 3 π + nπ , where n is an integer. Therefore, we have 4 3 π − 2 x = 4 3 π + nπ
Solving for x Solving for x , we get: − 2 x = nπ x = − 2 nπ
Finding Valid n Since x ∈ [ 0 , π ] , we need to find the integer values of n such that 0 ≤ − 2 nπ ≤ π . This implies 0 ≥ n ≥ − 2 . Thus, n can be 0 , − 1 , − 2 .
Calculating x Values For n = 0 , x = − 2 0 ⋅ π = 0 .
For n = − 1 , x = − 2 ( − 1 ) ⋅ π = 2 π .
For n = − 2 , x = − 2 ( − 2 ) ⋅ π = π .
Solution Set Therefore, the solution set is { 0 , 2 π , π } .
Examples
Imagine you're designing a robotic arm that needs to reach a specific point at a certain angle. The equation tan ( 4 3 π − 2 x ) = − 1 can model the arm's required angle adjustments. By solving this equation, you determine the precise angles the arm needs to rotate to achieve the desired position. This ensures the robot performs its task accurately, whether it's assembling components or performing surgery. Understanding trigonometric equations helps in programming precise movements in robotics and automation.
The solutions to the equation tan ( 4 3 π − 2 x ) = − 1 within the interval [ 0 , π ] are { 0 , 2 π , π } . Thus, the correct answer is option B: { 0 , 2 π , π } .
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