An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate.
Find All solutions of the equation
2sin(theta/3)+ square root three =0
2sin θ/3 + √3 = 0
sin θ/3 = -√3/2
θ/3 = 4π/3 or 5π/3 + 2nπ
...
To find all the solutions of the equation 2sin(theta/3) + square root three = 0, we can use algebraic methods. Here's how you can solve it:
Step 1: Subtract square root three from both sides of the equation to isolate the sine term:
2sin(theta/3) = -square root three
Step 2: Divide both sides of the equation by 2 to solve for the sine term:
sin(theta/3) = -square root three / 2
Step 3: Since we're looking for all solutions, we can find the general solution by using the inverse trigonometric function arcsin:
theta/3 = arcsin(-square root three / 2)
Step 4: To find the value of theta, we need to multiply both sides of the equation by 3:
theta = 3 * arcsin(-square root three / 2)
Step 5: Now, we can use a calculator to find the value of the arcsin(-square root three / 2). Let's denote this value as x:
x = arcsin(-square root three / 2)
Step 6: Next, we can substitute the value of x into the equation for theta:
theta = 3 * x
Step 7: Finally, we can find all the solutions for theta by substituting different values of k (integer) into the equation for theta, because the sine function is periodic:
theta = 3 * x + 6 * pi * k, where k is any integer.
So, the solutions to the equation 2sin(theta/3) + square root three = 0 are:
theta = 3 * x + 6 * pi * k, where x is the value of arcsin(-square root three / 2) and k is any integer.
To find all the solutions of the equation 2sin(theta/3) + sqrt(3) = 0, we can follow these steps:
Step 1: Subtract sqrt(3) from both sides of the equation:
2sin(theta/3) = -sqrt(3)
Step 2: Divide both sides of the equation by 2:
sin(theta/3) = -sqrt(3)/2
Step 3: Find the reference angle in the unit circle for sin(theta/3) = -sqrt(3)/2.
The reference angle for this value of sin(theta/3) can be found by looking for the angle whose sine is positive and equal to sqrt(3)/2. This angle is pi/3.
Step 4: Determine the general solutions.
The general solutions for sin(theta/3) = -sqrt(3)/2 can be found using the periodicity of the sine function. Since sin(theta/3) has a periodicity of 2pi, we can write the general solutions as:
theta/3 = pi*k + (-1)^k * pi/3, where k is any integer.
Step 5: Multiply both sides of the equation by 3 to solve for theta:
theta = 3(pi*k + (-1)^k * pi/3)
Therefore, the solutions for the equation 2sin(theta/3) + sqrt(3) = 0 are:
theta = 3(pi*k + (-1)^k * pi/3), where k is any integer.