Solve the equation on the interval 0<theta<2pi
-tan(2theta)+2sin(theta)=0
To solve the given equation on the interval 0 < θ < 2π, we will follow these steps:
Step 1: Rearrange the equation by using the identity tan(2θ) = 2tan(θ)/(1-tan^2(θ)).
-tan(2θ) + 2sin(θ) = 0
-(2tan(θ)/(1-tan^2(θ))) + 2sin(θ) = 0
Step 2: Multiply the entire equation by (1 - tan^2(θ)) to eliminate the denominator.
-(2tan(θ)) + 2sin(θ)(1 - tan^2(θ)) = 0
Step 3: Distribute and simplify the equation.
-2tan(θ) + 2sin(θ) - 2sin(θ)tan^2(θ) = 0
Step 4: Factor out a common factor.
2(tan(θ) - sin(θ)(1 + tan^2(θ))) = 0
Step 5: Set each factor equal to zero and solve for θ.
tan(θ) - sin(θ)(1 + tan^2(θ)) = 0
tan(θ) = sin(θ)(1 + tan^2(θ))
Now, let's solve each factor separately:
Factor 1: tan(θ) = 0
From 0 < θ < 2π, the solutions are θ = 0 and π.
Factor 2: sin(θ)(1 + tan^2(θ)) = 0
Using the Zero Product Property, we have two cases:
Case 1: sin(θ) = 0
θ = 0, π, and 2π (since 0 < θ < 2π).
Case 2: 1 + tan^2(θ) = 0
tan^2(θ) = -1
There are no real solutions for this case since tan^2(θ) is always greater than or equal to 0.
In conclusion, the solutions in the interval 0 < θ < 2π for the equation -tan(2θ) + 2sin(θ) = 0 are:
θ = 0, π, and 2π.
To solve the equation -tan(2θ) + 2sin(θ) = 0 on the interval 0 < θ < 2π, we can follow these steps:
Step 1: Simplify the equation using trigonometric identities.
-tan(2θ) + 2sin(θ) = 0
We can rewrite tan(2θ) as sin(2θ) / cos(2θ) using the identity tan(x) = sin(x) / cos(x).
-sin(2θ) / cos(2θ) + 2sin(θ) = 0
To add or subtract terms, we need them to have the same denominator. Multiply the second term by cos(2θ) / cos(2θ).
-sin(2θ) + 2sin(θ)cos(2θ) = 0
Step 2: Factor out sin(θ) from the equation.
sin(θ)(-cos(2θ) + 2cos(2θ)) = 0
sin(θ)(cos(2θ)) = 0
Step 3: Set each factor equal to zero and solve individually.
sin(θ) = 0
We know that sin(0) = 0, so θ = 0 is one solution.
Also, sin(π) = 0, so θ = π is another solution.
cos(2θ) = 0
Set 2θ equal to 90 degrees or π/2 radians since cos(90°) = 0.
So, 2θ = π/2 + 2nπ or 2θ = 3π/2 + 2nπ, where n is an integer.
Divide both sides by 2 to solve for θ.
θ = π/4 + nπ or θ = 3π/4 + nπ, where n is an integer.
Step 4: Determine the values of θ that are within the interval 0 < θ < 2π.
From the previous steps, we have the following solutions:
θ = 0, π, π/4, π/4 + π, 3π/4, 3π/4 + π
However, we need to make sure they fall within the interval 0 < θ < 2π.
The solutions θ = 0 and θ = π are already within the interval.
For θ = π/4 and θ = 3π/4 + π, we can simplify them:
θ = π/4 + π = 5π/4
θ = 3π/4 + π = 7π/4
So, the solutions within the interval 0 < θ < 2π are:
θ = 0, π, π/4, 5π/4, 3π/4, 7π/4