Find all solutions to the equation
tan(theta)+sqrt3=0
Write your answer in terms of pi and us the or button as necessary.
tan Ø = -√3
I know that tan 60 = +√3
and Ø could be in II or in IV
in II , Ø = 180-60° = 120° or 2π/3
in IV, Ø = 360-60 = 300° or 5π/3
To find all solutions to the equation tan(theta) + sqrt(3) = 0, we first need to isolate the tangent function and find the values of theta that satisfy the equation.
Starting with the equation: tan(theta) + sqrt(3) = 0
Step 1: Subtract sqrt(3) from both sides of the equation:
tan(theta) = -sqrt(3)
Step 2: Take the inverse tangent (arctan) of both sides to find the values of theta:
theta = arctan(-sqrt(3))
Now we have theta = arctan(-sqrt(3)). However, arctan alone does not give us the complete solution set. Since the tangent function has a periodicity of pi, we need to find all possible angles that satisfy the equation.
Step 3: Determine the general solution:
tan(theta) = -sqrt(3)
theta = arctan(-sqrt(3)) + n*pi, where n is an integer
By adding n*pi to the arctan value, we account for the periodicity of the tangent function and obtain the general solution set.
In summary, the solutions to the equation tan(theta) + sqrt(3) = 0 in terms of pi are:
theta = arctan(-sqrt(3)) + n*pi, where n is an integer.