Problem

Use the graph pair to map \(\text{sine}\) features to \(\text{cosecant}\). State the reciprocal identity, \(\text{sine}\) zero set, \(\text{cosecant}\) asymptotes, \(\text{sine}-\text{extrema}-\text{to}-\text{cosecant}-\text{vertex}\) mapping, \(\text{cosecant}\) zeros, and \(\text{cosecant}\) range.

Exact sine and cosecant overlay showing source zeros, repeated reciprocal asymptotes, and coincident plus-or-minus-one vertices.

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csc(x) = 1/sin(x). The source zero set is sine zeros x = k*pi, so the reciprocal asymptotes are cosecant vertical asymptotes x = k*pi. The source feature sine extrema at (pi/2 + 2k*pi, 1) and (3pi/2 + 2k*pi, -1) maps to cosecant vertices at those same points. Cosecant has zeros x = k*pi and its range is all real numbers.

csc(x) = sin(x). The source zero set is sine zeros x = k*pi, so the reciprocal asymptotes are cosecant vertical asymptotes x = k*pi. The source feature sine extrema at (pi/2 + 2k*pi, 1) and (3pi/2 + 2k*pi, -1) maps to cosecant vertices at those same points. Cosecant has no zeros and its range is y <= -1 or y >= 1.

csc(x) = 1/sin(x). The source zero set is sine zeros x = k*pi, so the reciprocal asymptotes are cosecant vertical asymptotes x = pi/2 + k*pi. The source feature sine extrema at (pi/2 + 2k*pi, 1) and (3pi/2 + 2k*pi, -1) maps to cosecant vertices at those same points. Cosecant has no zeros and its range is y <= -1 or y >= 1.

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