What are all the solutions for 4sintheta + 1 = 2sintheta from 0 to 2pi?
Would it be 7pi/6 and 11pi/6?
4 sin ( theta ) + 1 = 2 sin ( theta ) Subtract 2 sin ( theta to both sides )
4 sin ( theta ) + 1 - 2 sin ( theta ) = 2 sin ( theta ) - 2 sin ( theta )
2 sin ( theta ) + 1 = 0 Subtract 1 to both sides
2 sin ( theta ) + 1 - 1 = 0 - 1
2 sin ( theta ) = - 1 Divide both sides by 2
2 sin ( theta ) / 2 = - 1 / 2
sin ( theta ) = - 1 / 2
Your answer is correct.
theta = 7 pi / 6
and
theta = 11 pi / 6
To find all the solutions for the equation 4sinθ + 1 = 2sinθ from 0 to 2π, we can start by simplifying the equation:
4sinθ + 1 = 2sinθ
Rearrange the equation to isolate the sinθ term:
4sinθ - 2sinθ = -1
Combine like terms:
2sinθ = -1
Divide both sides by 2:
sinθ = -1/2
Now, we need to determine the values of θ that satisfy this equation between 0 and 2π.
To find the solutions, we can refer to the unit circle or the trigonometric ratios of common angles.
On the unit circle, the sine of an angle is negative in the third and fourth quadrants.
The angle whose sine value is -1/2 in the third quadrant is (7π/6).
And the angle whose sine value is -1/2 in the fourth quadrant is (11π/6).
Therefore, the solutions for the equation 4sinθ + 1 = 2sinθ from 0 to 2π are:
θ = 7π/6 and θ = 11π/6.
So, your initial response was correct. The solutions are indeed 7π/6 and 11π/6.
To find the solutions for the equation 4sin(theta) + 1 = 2sin(theta) in the given interval [0, 2pi], we need to manipulate the equation and solve for theta.
Step 1: Start with the given equation:
4sin(theta) + 1 = 2sin(theta)
Step 2: Simplify the equation by subtracting 2sin(theta) from both sides:
2sin(theta) + 1 = 0
Step 3: Subtract 1 from both sides to isolate the sine term:
2sin(theta) = -1
Step 4: Divide both sides by 2 to solve for sin(theta):
sin(theta) = -1/2
Step 5: To find the solutions for sin(theta) = -1/2, we need to refer to the unit circle and determine the angles where sin(theta) equals -1/2.
From the unit circle, we know that sin(theta) = -1/2 at two different angles: -π/6 and -5π/6.
Step 6: To find the solutions in the interval [0, 2pi], we need to find the corresponding positive angles.
The positive angle that corresponds to -π/6 is obtained by adding 2π to it:
Positive angle: -π/6 + 2π = 11π/6
The positive angle that corresponds to -5π/6 is obtained by adding 2π to it:
Positive angle: -5π/6 + 2π = 7π/6
So, the solutions for the equation 4sin(theta) + 1 = 2sin(theta) in the interval [0, 2pi] are theta = 7π/6 and theta = 11π/6.
Therefore, you are correct. The solutions for the equation in the specified interval are 7π/6 and 11π/6.