Problem

Use the graph pair to map \(\text{tangent}\) features to \(\text{cotangent}\). State \(\text{cotangent}\)'s \(\text{sine}-\text{cosine}\) quotient, how \(\text{tangent}\) zeros map to \(\text{cotangent}\) asymptotes, how \(\text{tangent}\) asymptotes map to \(\text{cotangent}\) zeros, and \(\text{cotangent}\)'s range.

Exact tangent and cotangent overlay showing interlaced zeros and vertical asymptotes over a symmetric two-pi interval.

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cot(x) = cos(x)/sin(x). The source zero set is tangent zeros x = k*pi, where sin(x) = 0, so the reciprocal asymptotes are cotangent vertical asymptotes x = pi/2 + k*pi. The source feature tangent asymptotes x = pi/2 + k*pi, where cos(x) = 0 and sin(x) is nonzero maps to cotangent zeros x = pi/2 + k*pi. Cotangent has all real values; the cos/sin form correctly handles tangent's undefined points.

cot(x) = cos(x)/sin(x). The source zero set is tangent zeros x = k*pi, where sin(x) = 0, so the reciprocal asymptotes are cotangent vertical asymptotes x = k*pi. The source feature tangent asymptotes x = pi/2 + k*pi, where cos(x) = 0 and sin(x) is nonzero maps to cotangent zeros x = pi/2 + k*pi. Cotangent has range [-1, 1]; the cos/sin form correctly handles tangent's undefined points.

cot(x) = tan(x). The source zero set is tangent zeros x = k*pi, where sin(x) = 0, so the reciprocal asymptotes are cotangent vertical asymptotes x = k*pi. The source feature tangent asymptotes x = pi/2 + k*pi, where cos(x) = 0 and sin(x) is nonzero maps to cotangent zeros x = pi/2 + k*pi. Cotangent has all real values; the cos/sin form correctly handles tangent's undefined points.

cot(x) = cos(x)/sin(x). The source zero set is tangent zeros x = k*pi, where sin(x) = 0, so the reciprocal asymptotes are cotangent vertical asymptotes x = k*pi. The source feature tangent asymptotes x = pi/2 + k*pi, where cos(x) = 0 and sin(x) is nonzero maps to cotangent zeros x = pi/2 + k*pi. Cotangent has all real values; the cos/sin form correctly handles tangent's undefined points.

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