Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer.

tan θ = − sqrt 3/3

tan(?)=-1/3

?=

To solve the equation tan θ = -sqrt(3)/3, we need to find the values of θ.

Step 1: Find the reference angle.
The reference angle can be found using the equation tan θ = opposite/adjacent.
In this case, we have tan θ = -sqrt(3)/3.
Since tan θ is negative, the angle θ lies in the 2nd or 4th quadrant.
To find the reference angle, we take the absolute value of the ratio sqrt(3)/3, which is sqrt(3)/3.

Step 2: Find the 2 possible angles.
Since the angle θ can lie in the 2nd or 4th quadrant, we need to find the 2 possible angles.
We can find these angles using the inverse tangent function on the reference angle.
In this case, the possible angles are θ = arctan(sqrt(3)/3) and θ = arctan(sqrt(3)/3) + π.

Step 3: Simplify the angles.
Using a calculator, we can find the values of the angles:
θ = arctan(sqrt(3)/3) ≈ 30 degrees ≈ π/6 radians.
θ = arctan(sqrt(3)/3) + π ≈ 210 degrees ≈ 7π/6 radians.

Therefore, the solutions to the equation are θ ≈ π/6 and θ ≈ 7π/6.
So, the final answer is π/6, 7π/6 (as a comma-separated list).

To solve the given equation tan θ = -√3/3, we can use the inverse tangent or arctan function to find the value of θ.

Step 1: Use the inverse tangent function (tan^(-1)) to find the angle whose tangent is -√3/3.

θ = tan^(-1)(-√3/3)

Step 2: Evaluate the inverse tangent expression.

θ ≈ -30° or -π/6 (in radians)

Step 3: Since the tangent function is periodic, we can find additional solutions by adding or subtracting multiples of 180° (or π radians) from the initial solution.

θ = -30° + 180°k, where k is any integer

In radians, θ = -π/6 + πk, where k is any integer

Therefore, the general solutions to the equation tan θ = -√3/3 are:

θ ≈ -30° + 180°k, or θ ≈ -π/6 + πk, where k is any integer.

tanØ = -√3/3

from the 30-60-90 triangle, you should recognize that tan 30° = 1/√3 = √3/3

or , using your calculator find tan^1 (√3/3) to get 30°

but the tangent is negative in II and Iv
so Ø = 150°
or Ø = 330°

check with your calculator, it is correct