Problem

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

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

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What this problem is really about

A reciprocal graph is best traced from a few decisive source values. Source zeros make the reciprocal undefined and become vertical asymptotes, while source values of positive or negative one stay at the same height as branch vertices. Between those landmarks, reciprocal magnitude explains the branch shape, the absence of zeros, and the separated range.

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sec(x) = 1/cos(x). The source zero set is cosine zeros x = pi/2 + k*pi, so the reciprocal asymptotes are secant vertical asymptotes x = pi/2 + k*pi. The source feature cosine extrema (k*pi, (-1)k) maps to secant vertices (k*pi, (-1)k). Secant has zeros x = pi/2 + k*pi and its range is all real numbers.

sec(x) = cos(x). The source zero set is cosine zeros x = pi/2 + k*pi, so the reciprocal asymptotes are secant vertical asymptotes x = pi/2 + k*pi. The source feature cosine extrema (k*pi, (-1)k) maps to secant vertices (k*pi, (-1)k). Secant has no zeros and its range is y <= -1 or y >= 1.

sec(x) = 1/cos(x). The source zero set is cosine zeros x = pi/2 + k*pi, so the reciprocal asymptotes are secant vertical asymptotes x = k*pi. The source feature cosine extrema (k*pi, (-1)k) maps to secant vertices (k*pi, (-1)k). Secant has no zeros and its range is y <= -1 or y >= 1.

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

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