Solve the equation on the interval [tex]$0 \leq 0\ \textless \ 2 \pi$[/tex].
[tex]$(\tan \theta+1)(\cos \theta-1)=0$[/tex]
Answer (1)
Set each factor to zero: tan θ + 1 = 0 or cos θ − 1 = 0 .
Solve tan θ = − 1 , which gives θ = 4 3 π and θ = 4 7 π .
Solve cos θ = 1 , which gives θ = 0 .
The solutions are 0 , 4 3 π , 4 7 π .
Explanation
Understanding the Problem We are given the equation ( tan θ + 1 ) ( cos θ − 1 ) = 0 and asked to solve it on the interval 0 ≤ θ < 2 π . This means we need to find all values of θ within this interval that satisfy the equation.
Setting Factors to Zero To solve the equation, we set each factor equal to zero: tan θ + 1 = 0 or cos θ − 1 = 0
Solving tan(θ) + 1 = 0 First, let's solve tan θ + 1 = 0 . This is equivalent to tan θ = − 1 . The tangent function is negative in the second and fourth quadrants. The reference angle for tan − 1 ( 1 ) is 4 π . Therefore, the solutions in the interval [ 0 , 2 π ) are θ = π − 4 π = 4 3 π and θ = 2 π − 4 π = 4 7 π
Solving cos(θ) - 1 = 0 Next, let's solve cos θ − 1 = 0 . This is equivalent to cos θ = 1 . The cosine function is equal to 1 at 0 and 2 π . However, since the interval is 0 ≤ θ < 2 π , we only include 0 . Thus, the solution is θ = 0
Combining Solutions Combining the solutions from both equations, we have θ = 0 , 4 3 π , 4 7 π These are all the solutions in the interval 0 ≤ θ < 2 π .
Final Answer Therefore, the solutions to the equation ( tan θ + 1 ) ( cos θ − 1 ) = 0 on the interval 0 ≤ θ < 2 π are 0 , 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 rotate to achieve those positions. For example, if the arm's movement is described by trigonometric functions, finding the solutions to these equations tells you the precise angles required to position the arm correctly. This is crucial in robotics, engineering, and any field where precise angular movements are necessary.
