Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$2 \cos \theta-1=0$[/tex]
What are the solutions to [tex]$2 \cos \theta-1=0$[/tex] 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 your answer in the form of a comma separated expression. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Isolate cos θ in the equation: 2 cos θ − 1 = 0 ⟹ cos θ = 2 1 .
Find the angles θ in the interval [ 0 , 2 π ) such that cos θ = 2 1 .
The solutions are θ = 3 π and θ = 3 5 π .
The solution set is { 3 π , 3 5 π } .
Explanation
Understanding the Problem We are asked to solve the equation 2 cos θ − 1 = 0 for θ in the interval 0 ≤ θ < 2 π . This means we need to find all angles θ between 0 and 2 π (not including 2 π ) whose cosine is a specific value.
Isolating cos θ First, we isolate cos θ in the given equation: 2 cos θ − 1 = 0 Add 1 to both sides: 2 cos θ = 1 Divide both sides by 2: cos θ = 2 1
Finding the Angles Now we need to find the angles θ in the interval [ 0 , 2 π ) such that cos θ = 2 1 . We know that cos 3 π = 2 1 . Since cosine is positive in the first and fourth quadrants, we have two solutions in the interval [ 0 , 2 π ) .
The first solution is θ = 3 π , which is in the first quadrant.
The second solution is in the fourth quadrant. To find this angle, we subtract 3 π from 2 π : θ = 2 π − 3 π = 3 6 π − 3 π = 3 5 π
Final Solutions Therefore, the solutions to the equation 2 cos θ − 1 = 0 in the interval 0 ≤ θ < 2 π are θ = 3 π and θ = 3 5 π .
Examples
Understanding trigonometric equations is crucial in many fields, such as physics and engineering. For example, 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 the times at which the pendulum reaches specific positions or velocities. Similarly, in electrical engineering, alternating current (AC) circuits involve sinusoidal functions, and solving trigonometric equations is essential for analyzing circuit behavior and designing filters.
