Solve for theta.
6 sin theta-1=2 sin theta +1
0 ≤ theta < 2 pi
Please show the work as equations. Give the exact radian measure and no decimals.
We can solve this equation by isolating the sine function on one side and using trigonometric identities to simplify it:
6 sin theta - 1 = 2 sin theta + 1
4 sin theta = 2
sin theta = 1/2
This means that theta could be in either the first or second quadrant (where sine is positive), and the reference angle for theta is pi/6. Therefore, the solutions are:
theta = pi/6 or theta = 5pi/6
If we add multiples of 2pi to these values, we can get all possible solutions in the given interval of 0 ≤ theta < 2 pi. Therefore, the solutions are:
theta = pi/6 + 2npi or theta = 5pi/6 + 2npi, where n is an integer.
To solve for theta in the equation 6 sin(theta) - 1 = 2 sin(theta) + 1, we can start by gathering all the terms involving sin(theta) on one side of the equation and the constant terms on the other side. Here's how to proceed:
6 sin(theta) - 2 sin(theta) = 1 + 1 + 0
(6 - 2) sin(theta) = 2
Now, simplify the equation:
4 sin(theta) = 2
Next, isolate sin(theta) by dividing both sides of the equation by 4:
sin(theta) = 2/4
sin(theta) = 1/2
To find the values of theta that satisfy this equation, we can use the inverse sine function or the unit circle. Since the given domain is 0 ≤ theta < 2 pi, we will use the unit circle approach.
Recall that sin(theta) = opposite/hypotenuse in a right triangle. For a given angle, the opposite side of the triangle on the unit circle is the y-coordinate, while the hypotenuse is always 1.
The unit circle helps us find the angles where sin(theta) = 1/2. From the unit circle, the angles that satisfy this condition are π/6 and 5π/6.
So, in radians, the solutions for theta are:
θ = π/6 and θ = 5π/6
To solve the equation 6sin(theta) - 1 = 2sin(theta) + 1, we want to isolate sin(theta) terms on one side of the equation.
First, let's move the 2sin(theta) term to the left side and the -1 term to the right side by adding 1 to both sides of the equation:
6sin(theta) - 2sin(theta) = 1 + 1
This simplifies to:
4sin(theta) = 2
Next, divide both sides of the equation by 4 to solve for sin(theta):
sin(theta) = 2/4
Simplifying further:
sin(theta) = 1/2
To find the value of theta, we need to find the inverse sine (also known as arcsine) of 1/2.
Using a reference triangle, we know that sin(theta) = 1/2 when theta = pi/6 or 30 degrees.
Since the given range for theta is 0 ≤ theta < 2pi, we need to find all values of theta that satisfy this equation. Recall that the sine function has a period of 2pi, meaning it repeats every 2pi radians.
So, the solutions for theta are:
theta = pi/6 + 2pi(n), where n is an integer and n ≥ 0.
Therefore, the exact radian measures for theta within the given range are:
theta = pi/6, 7pi/6, 13pi/6, and 19pi/6.