Find all angles in degrees that satisfy the equationtan∝ +√3=0
120, 240, and 300
that is, tan ∝ = -√3
in QI, tan pi/3 = √3
so, you want an angle in QII or QIV
pi-pi/3 = 2pi/3
or
2pi-pi/3 = 5pi/3
To find the values of α that satisfy the equation tan(α) + √3 = 0, we can start by isolating the tangent term:
tan(α) = -√3
Next, we can take the inverse tangent (also called arctan or tan^(-1)) of both sides to find the values of α:
α = arctan(-√3)
To find the complete solution, we need to consider the range of the arctan function. The inverse tangent function has a range of -π/2 to π/2 (-90° to 90°). However, since we're looking for solutions that make tan(α) + √3 = 0, the angles should fall within the range where the tangent is negative.
By observing the unit circle, we can determine that for α in the range between -π and 0, the tangent function is negative. So, we need to find the values of α that fall within these bounds.
Using a scientific calculator or trigonometric tables, we can find the principal angle whose tangent is -√3:
α = arctan(-√3) ≈ -60°
Since the tangent function has a periodicity of π (180°), we can add or subtract multiples of π to find additional solutions. Adding π:
α = -60° + 180° = 120°
Therefore, the angles that satisfy the equation tan(α) + √3 = 0 are -60° and 120°.