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.)
(tan^2θ-9)(2cosθ+1)=0
θ=__________________
I got 2π/3+2πn, πn+1.249, 4π/3+2πn, πn-1.249
It was wrong :(
so, you got
tanθ = ±√3
cosθ = -1/2
tanθ = √3 means θ = π/3 or 4π/3
tanθ = -√3 means θ = 2π/3 or 5π/3
cosθ = -1/2 means θ = 2π/3 or 4π/3
so, that gives
nπ + π/3
nπ + 2π/3
nπ + 4π/3
nπ + 5π/3
2nπ + 2π/3
2nπ + 4π/3
That all boils down to just
nπ + 2π/3
nπ + 4π/3
To solve the equation (tan^2θ-9)(2cosθ+1)=0, we can set each factor equal to zero and solve for θ.
For the first factor, tan^2θ-9=0, we can use the identity tan^2θ = sec^2θ - 1. Therefore, we have sec^2θ - 1 - 9 = 0.
Combining like terms, we get sec^2θ - 10 = 0. Rearranging the equation, we have sec^2θ = 10.
Taking the square root of both sides, we get secθ = ±√10. To find θ, we need to take the inverse secant (or arcsec) of both sides.
Now, the second factor is 2cosθ+1=0. Solving this equation, we subtract 1 from both sides, giving us 2cosθ = -1. Dividing by 2, we have cosθ = -1/2. To find θ, we take the inverse cosine (or arccos) of both sides.
Let's calculate the solutions for each factor.
For the first factor, secθ = ±√10, we take the inverse secant of ±√10. Using a calculator in degrees mode, we find that the solution in degrees is approximately 15.639° and 164.361°. However, we need to consider that the equation involves periodic functions (trigonometric functions), which repeat after certain intervals (periods).
For the second factor, cosθ = -1/2, we take the inverse cosine of -1/2. Using a calculator in degrees mode, we find that the solution in degrees is approximately 120° and 240°.
Now, let's put it all together.
The solutions for the equation (tan^2θ-9)(2cosθ+1)=0 are:
θ = 15.639° + 180°k, where k is an integer (for the first factor)
θ = 164.361° + 180°k, where k is an integer (for the first factor)
θ = 120° + 360°n, where n is an integer (for the second factor)
θ = 240° + 360°n, where n is an integer (for the second factor)
Please note that when using radians instead of degrees, the values will be different, but the concept and steps remain the same.