Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
sin θ cos θ − 2 sin θ = 0
sin θ cos θ − 2 sin θ = 0
factor out sin(θ)
sin(θ)(cos(θ)-2) = 0
Can you take it from here?
that is where i got stuck
When a product of factors is zero, one of the factors must be zero.
So, either
sin(θ) = 0 or cos(θ)-2 = 0
sin(θ) = 0 when θ = k*pi
cos(θ)-2 = 0 when cos(θ) = 2
but cos(θ) is always less than 1. No solution here.
To solve the equation sin θ cos θ − 2 sin θ = 0, we can factor out sin θ from the equation.
sin θ(cos θ - 2) = 0
Setting each factor equal to zero gives us two equations:
sin θ = 0 .......(1)
cos θ - 2 = 0 .......(2)
To solve equation (1), we need to find the values of θ where sin θ equals zero. Recall that the sine function is zero at 0, π, 2π, and so on. However, since we have parameter k, we can express the general solution as θ = kπ, where k is an integer.
Now, let's solve equation (2) for cos θ:
cos θ - 2 = 0
cos θ = 2
The cosine function only takes values between -1 and 1. Therefore, there is no real number solution for θ that satisfies this equation.
Thus, the solution to the given equation sin θ cos θ − 2 sin θ = 0, is θ = kπ, where k is an integer.