Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$\cos \left(2 \theta-\frac{\pi}{2}\right)=1$[/tex]
What are the solutions in the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex]? Select the correct choice.
A. The solution set is { }. (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. Type an expression. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Find the general solution to cos ( x ) = 1 , which is x = 2 nπ .
Set 2 θ − 2 π = 2 nπ and solve for θ , obtaining θ = nπ + 4 π .
Find the values of n such that 0 ≤ θ < 2 π .
The solutions are θ = 4 π and θ = 4 5 π , so the answer is 4 π , 4 5 π .
Explanation
Understanding the Problem We are asked to solve the equation cos ( 2 θ − 2 π ) = 1 for θ in the interval 0 ≤ θ < 2 π . In other words, we need to find all angles θ between 0 and 2 π that satisfy the given equation.
General Solution The general solution to cos ( x ) = 1 is x = 2 nπ , where n is an integer. This means that the cosine function equals 1 at integer multiples of 2 π .
Solving for Theta Therefore, we must have 2 θ − 2 π = 2 nπ for some integer n . Now we solve for θ :
Add 2 π to both sides: 2 θ = 2 nπ + 2 π Divide both sides by 2: θ = nπ + 4 π This gives us the general solution for θ .
Finding Solutions in the Interval Now we need to find the values of n such that 0 ≤ θ < 2 π . We will test different integer values of n to find the solutions within the given interval.
For n = 0 :
θ = 0 ⋅ π + 4 π = 4 π Since 0 ≤ 4 π < 2 π , this is a valid solution.
For n = 1 :
θ = 1 ⋅ π + 4 π = π + 4 π = 4 5 π Since 0 ≤ 4 5 π < 2 π , this is also a valid solution.
For n = 2 :
θ = 2 ⋅ π + 4 π = 2 π + 4 π = 4 9 π Since 2\pi"> 4 9 π > 2 π , this solution is outside the given interval.
Final Solutions Therefore, the solutions in the interval 0 ≤ θ < 2 π are θ = 4 π and θ = 4 5 π .
Examples
Understanding trigonometric equations is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, you often encounter equations involving trigonometric functions. Solving these equations allows you to predict the position of the pendulum at a given time or to determine the current flow in the circuit. This skill is also essential in music synthesis, where trigonometric functions are used to create and manipulate sound waves.
