For what values of θ is the tangent 0? For what values of θ is the tangent undefined? Explain.
The tangent of an angle θ is defined as the ratio of the sine of θ to the cosine of θ, and is written as tan(θ) = sin(θ) / cos(θ).
The tangent function is equal to 0 when the numerator, sin(θ), is equal to 0. This occurs when θ is a multiple of π (i.e. θ = nπ where n is an integer), because sin(θ) is 0 at these angles. Therefore, the values of θ for which the tangent is 0 are θ = nπ where n is an integer.
The tangent function is undefined when the denominator, cos(θ), is equal to 0. This occurs when θ is equal to (2n + 1)π/2 where n is an integer, because cos(θ) is 0 at these angles. At these angles, dividing by 0 is undefined, so the tangent function is undefined. Therefore, the values of θ for which the tangent is undefined are θ = (2n + 1)π/2 where n is an integer.