Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , that solve the following equation.tangent, theta, equals, minus, start fraction, square root of, 3, end square root, divided by, 3, end fractiontanθ=− 33
Question
Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , that solve the following equation.tangent, theta, equals, minus, start fraction, square root of, 3, end square root, divided by, 3, end fractiontanθ=− 33
Solution
The given equation is tan(θ) = -√3/3.
First, we need to find the reference angle. The reference angle is the acute angle made with the x-axis.
The reference angle for tan(θ) = √3/3 is 30 degrees because tan(30) = √3/3.
However, since the tangent is negative, we are dealing with either the second or the fourth quadrant (as in these quadrants, the tangent function is negative).
In the second quadrant, the angle is 180 - reference angle = 180 - 30 = 150 degrees.
In the fourth quadrant, the angle is 360 - reference angle = 360 - 30 = 330 degrees.
So, the solutions to the equation tan(θ) = -√3/3 in the interval 0 ≤ θ < 360 are θ = 150 degrees and θ = 330 degrees.
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