
Trigonometry 



Trigonometric
Functions 
Signs of trigonometric
functions 
The table of signs of trigonometric
functions 





Signs of trigonometric
functions

Quadrant I
 Values of trigonometric functions, sine, cosine, tangent and cotangent of any arc from
the first quadrant are all positive as positive are the coordinates of the points,
P,
S_{1}
and S_{2}
that define their values.


Quadrant II
 For arcs from the second quadrant points, P
and S_{2}
both have negative abscissas (see the above figure), so the cosine and cotangent are negative. The ordinate of the terminal point
P
is positive so that the sine is positive while the ordinate of the point
S_{1}
is negative, thus the tangent is negative.


Quadrant III
 As the abscissas and the ordinates of the terminal points P
of arcs from the third quadrant (see the above figure) are negative it follows that cosine and sine functions of these arcs are negative. The ordinates
of the points
S_{1} and the abscissas of the points
S_{2} that belong to the arcs from the third quadrant are positive. Thus, the tangent and cotangent of these arcs are positive.


Quadrant IV
 The functions, sine, tangent and cotangent of the arcs from the fourth quadrant are negative as are the coordinates of the points,
P,
S_{1}
and S_{2}, that belong to them. Only the cosine function of arcs from the fourth quadrant is positive as are the abscissas of points
P that belong to them
(see the above figure).


The table of signs of trigonometric
functions



Example:
To which quadrant belongs the endpoint of an arc
a
if
sin a < 0 and
cot a
> 0.

Solution: The
right figure shows that the ordinate of 
the endpoint P
of an arc from the third quadrant is 
negative, so sin
a < 0
while the abscissa of the

point (in which the extension of the terminal side of 
the angle
a
intersects the tangent y =
1) is positive, 
i.e., cot
a
> 0.

That is in accordance with the signs in the above

table.












Precalculus
contents C 



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