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J 54 TRIGONOMETRY.
Sec. 30° = 1.1547; that is, if the angle n is 30 degrees and
the line c gis 1 of any unit, the line cf is 1.1547 of the same unit.
Cosec. 30° = 2 ; that is, if the angle n is 30 degrees and the
line eg is 1 of any unit, the line eh is 2 of the same unit.
Tang. 30° = 0.57735 ; that is, if the angle n is 30 degrees
and the line c g is 1 of any unit, the line g f is 0.57735 of the
same unit.
Cot. 30° = 1.73205; that is, if the angle n is 30 degrees and
the line c g is 1 of any unit, the line k h is 1.73205 of the
same unit.
Increasing the angle n will increase sine, tangent and
secant, but will decrease cosine, cotangent and cosecant.
When the angle ;z approaches 90°, the tangents g f increase
more and more to infinite length. When n actually reaches 90°
of course c b coincides with c k and becomes parallel to gf, so
that in an angle of 90° both the secant and the tangent have
infinite length, which is denoted by the sign Oo ?
and cosine and
cotangent have vanished.
In the first quadrant (that is when angle n does not exceed
90°) the trigonometrical functions are all considered to be
positive and are denoted by -f (plus). When the angle n
exceeds 90°, only sine and cosecants remain positive ; all the
other functions have become negative and are denoted by —
(minus).
The following table gives the properties of the trigono-
metrical functions in the four different quadrants :
Degree. Sine. Cosine.
0° to 90°
90 °
to 180
°
180 ° to 270
°
270° to 360°
Increase from to radius +
Decrease from radius to -j-
Increase from to radius —
Decrease from radius to —
Decrease from radius to -j-
Increase from to radius —
Decrease from radius to —
Increase from to radius -f~
Degree. Secant. Cosecant.
0° to 90°
90° to 180°
180° to 270°
270° to 360°
Increase from radius to Co -)-
Decrease from cx> to radius —
Increase from radius to °° —
Decrease from 00 to radius +
Decrease from 0° to radius -f-
Increase from radius to °° -f~
Decrease from Co to radius —
Increase from radius to 0° —
Degree. Tangent. Cotangent.
0° to 90°
90 ° to 180
°
180° to 270°
270° to 360°
Increase from to 0° +
Decrease from 0° to —
Increase from to °° -f-
Decrease from 0° to —
Decrease from Oo to -f-
Increase from to 0° —
Decrease from °° to -j-
Increase from to °0 —
From the rule that the square of the hypothenuse is equal
to the sum of the squares of the base and the perpendicular, it
also follows that:
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