Deducing cos ⁡ θ &lt; sin ⁡ θ θ &lt; 1 from sin ⁡ θ &lt; θ &lt; sin ⁡ θ cos ⁡ θ

Question

Deducing   cos  ⁡  θ  &amp;lt;            sin      ⁡      θ        θ    &amp;lt;  1 from   sin  ⁡  θ  &amp;lt;  θ  &amp;lt;            sin      ⁡      θ              cos      ⁡      θ

nelppeazy9v3ie · Accepted Answer

The first part of the second line comes from the second part of the first line, and vice versa.
From

\sin θ < θ,

we get

\frac{\sin θ}{θ} < 1

.
From

θ < \frac{\sin θ}{\cos θ},

we get

\cos θ < \frac{\sin θ}{θ}

.

Lexi Chandler · Accepted Answer

Reverse  sin  ⁡  θ  &amp;lt;  θ  &amp;lt;            sin      ⁡      θ              cos      ⁡      θ      to get            cos      ⁡      θ              sin      ⁡      θ        &amp;lt;      1    θ    &amp;lt;      1          sin      ⁡      θ      Then, multiply   sin  ⁡  θ  &amp;gt;  0,  cos  ⁡  θ  &amp;lt;            sin      ⁡      θ        θ    &amp;lt;  1

Deducing cos ⁡ θ < sin

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