De-increasing trig ratios

Geometry Level 2

Consider the following expression,

N = sec ⁡ θ csc ⁡ θ , M = csc ⁡ θ sec ⁡ θ , θ ≥ 0 N = \frac{\sec \theta}{\csc \theta}, \qquad M = \frac{\csc \theta}{\sec \theta}, \qquad \theta \ge 0

Is it true that N > M N > M for some value of θ \theta ?

No Yes Inadequate information

Viki Zeta by Viki Zeta , 19, India

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1 solution

Tom Engelsman
Sep 20, 2018

We have N = t a n ( θ ) , M = c o t ( θ ) N = tan(\theta), M = cot(\theta) . One case where the inequality N > M N > M holds is θ ∈ ( π 4 , π 2 ) \theta \in (\frac{\pi}{4}, \frac{\pi}{2}) . So, the answer is YES for some values of θ ≥ 0. \theta \ge 0.

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