Plane trigonometry, by S.L. Loney.

26

TRIGONOMETRY.

Now sin20 and cosS0, being both squares, are both
necessarily positive. Hence, since their sum is unity,
neither of them can be greater than unity.
[For if one of them, say sin2 0, were greater than unity, the other,
cos2 8, would have to be negative, which is impossible.]
Hence neither the sine nor the cosine can be numerically greater than unity.
Since sin 0 cannot be greater than unity therefore
cosec 0, which equals si., cannot be numerically less
than unity.
So sec 0, which equals --, cannot be numerically
x cos 0'
less than unity.
30. The foregoing results follow easily from the figure
of Art. 23.
For, whatever be the value of the angle A OP,
neither the side OM nor the side MP is ever greater
than OP.
MP
Since MP is never greater than OP the ratio -p — is
never greater than unity, so that the sine of an angle is
never greater than unity.
Also since OM is never greater than OP, the ratio 0
is never greater than unity, i.e. the cosine is never greater
than unity.
31. We can express the trigonometrical ratios of an
angle in terms of any one of them.