Question

If θ = 5π/7 and β = -2π/7, do the tangents of these angles, i.e., tan(θ) and tan(β), have the same value?

Answer 1

The tangents of the angles θ = 5π/7 and β = -2π/7 do not have the same value. The tangent function's period and odd property indicate different values for these two angles.

Step-by-step explanation:

To determine if the tangents of the angles θ = 5π/7 and β = -2π/7 have the same value, we must understand the periodicity and symmetry properties of the tangent function. The tangent function has a period of π, meaning that tan(α) = tan(α + kπ) for any integer k. Additionally, we use the fact that tangent is an odd function: tan(-α) = -tan(α).

Applying these properties, we can evaluate:

• tan(5π/7) directly,
• tan(-2π/7) knowing that tan(β) = tan(-β) is the negative of tan(2π/7).

However, since 5π/7 and 2π/7 are not separated by an integer multiple of π, these angles do not lie on the same point of the tangent's period, and thus, their tangents are not equal. To add, tan(5π/7) is in the third quadrant where tangent is positive, while tan(-2π/7) which is equal to -tan(2π/7) is in the fourth quadrant where tangent is negative.