Solve the equation on the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex].
[tex]$2 \sin \theta+1=0$[/tex]
What are the solutions to [tex]$2 \sin \theta+1=0$[/tex] in the interval [tex]$0 \leq \theta\ \textless \ 2 \pi$[/tex]?
A. The solution set is $\square$ {As}
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as need expression. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Isolate sin θ : sin θ = − 2 1 .
Identify quadrants where sine is negative: third and fourth quadrants.
Find the reference angle: 6 π .
Determine the solutions in the specified interval: 6 7 π , 6 11 π .
The solution set is { 6 7 π , 6 11 π } .
Explanation
Understanding the Problem We are asked to solve the equation 2" , " s in θ + 1 = 0 on the interval 0" , " l e qθ < 2 π . This means we need to find all angles θ between 0 and 2 π (not including 2 π ) that satisfy the equation.
Isolating sin θ First, let's isolate sin θ in the equation: 2 sin θ + 1 = 0 2 sin θ = − 1 sin θ = − 2 1
Identifying Quadrants Now we need to find the angles θ in the interval [ 0 , 2 π ) for which sin θ = − 2 1 . Recall that sin θ is negative in the third and fourth quadrants.
Finding the Reference Angle The reference angle for sin θ = 2 1 is 6 π because sin 6 π = 2 1 .
Finding the Solutions Therefore, the solutions in the third and fourth quadrants are: Third quadrant: π + 6 π = 6 6 π + 6 π = 6 7 π Fourth quadrant: 2 π − 6 π = 6 12 π − 6 π = 6 11 π
Final Solutions So, the solutions to the equation 2 sin θ + 1 = 0 in the interval 0 ≤ θ < 2 π are θ = 6 7 π and θ = 6 11 π .
Examples
Imagine you're designing a Ferris wheel. The height of a rider above the ground can be modeled using a sine function. Solving trigonometric equations like the one above helps you determine at what angles the rider will be at a specific height. This is crucial for safety and ensuring a smooth ride, as it allows you to predict and control the wheel's position at any given time.
