Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$4 \sin ^2 \theta-1=0$[/tex]
What are the solutions in the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex]?
A. The solution set is { }. (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. expression. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Isolate sin 2 θ : sin 2 θ = 4 1 .
Take the square root: sin θ = ± 2 1 .
Find angles where sin θ = 2 1 : θ = 6 π , 6 5 π .
Find angles where sin θ = − 2 1 : θ = 6 7 π , 6 11 π .
The solution set is: 6 π , 6 5 π , 6 7 π , 6 11 π .
Explanation
Problem Analysis We are given the equation 4 sin 2 θ − 1 = 0 and we want to find all solutions for θ in the interval 0 ≤ θ < 2 π .
Isolating sin 2 θ First, we isolate sin 2 θ :
4 sin 2 θ = 1 sin 2 θ = 4 1
Taking the Square Root Next, we take the square root of both sides: sin θ = ± 4 1 sin θ = ± 2 1
Finding the Angles Now, we find the angles θ in the interval 0 ≤ θ < 2 π such that sin θ = 2 1 or sin θ = − 2 1 .
For sin θ = 2 1 , the solutions are θ = 6 π and θ = 6 5 π .
For sin θ = − 2 1 , the solutions are θ = 6 7 π and θ = 6 11 π .
Final Solutions Therefore, the solutions in the interval 0 ≤ θ < 2 π are 6 π , 6 5 π , 6 7 π , 6 11 π .
Examples
Understanding trigonometric equations is crucial in various fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the angle it makes with the vertical can be modeled using trigonometric functions. Solving equations like the one above helps determine specific angles at certain times, which is essential for predicting the pendulum's behavior. Similarly, in electrical engineering, alternating current (AC) circuits rely heavily on trigonometric functions to describe voltage and current variations. Finding solutions to trigonometric equations allows engineers to calculate key parameters like phase angles and power factors, ensuring efficient circuit design and operation.
