tan((theta/2)+(pi/6))=-1

To find the value of theta (θ) from the given equation tan((θ/2) + (π/6)) = -1, we can follow these steps:

Step 1: Identify the equation
The given equation is in the form of a trigonometric function (tan). We need to solve it for the variable θ.

Step 2: Use the tan addition formula
The tan addition formula states that tan(A + B) = (tan A + tan B) / (1 - tan A * tan B). In our equation, A = (θ/2) and B = π/6.

Applying the formula, we have:
tan((θ/2) + (π/6)) = (tan(θ/2) + tan(π/6)) / (1 - tan(θ/2) * tan(π/6)) = -1

Step 3: Substitute known values
We can substitute the known values for tan(π/6), which is 1/√3.

The equation becomes:
(tan(θ/2) + 1/√3) / (1 - tan(θ/2) * 1/√3) = -1

Step 4: Simplify the equation
To simplify further, we can multiply both sides of the equation by the denominator to eliminate it.

(tan(θ/2) + 1/√3) = -1 * (1 - tan(θ/2) * 1/√3)

(tan(θ/2) + 1/√3) = -1 + tan(θ/2)/√3

Multiplying both sides by √3 to eliminate the square root in the denominator:

√3 * (tan(θ/2) + 1/√3) = -√3 + (tan(θ/2) * √3) / √3

√3 * tan(θ/2) + 1 = -√3 + tan(θ/2)

Step 5: Isolate the variable
Move the terms with tan(θ/2) to one side and the constant terms to the other side:

√3 * tan(θ/2) - tan(θ/2) = -√3 - 1

(√3 - 1) * tan(θ/2) = -√3 - 1

Step 6: Solve for θ/2
Divide both sides of the equation by (√3 - 1):

tan(θ/2) = (-√3 - 1) / (√3 - 1)

Now, use a calculator or a trigonometric identity to find the value of tan(θ/2), which is approximately -2.7321.

Step 7: Find θ
Take the inverse tangent (arctan) of the value obtained in step 6 to find θ/2:

θ/2 = arctan(-2.7321)

Divide the result by 2 to get the value of θ:

θ = 2 * arctan(-2.7321)

Using a calculator, the approximate value of θ is approximately -4.761 radians or -273.13 degrees.