Trigonometric functions, signs of trigonometric functions

Trigonometry

Trigonometric Functions

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, S₁ and S₂ that define their values.

Quadrant II - For arcs from the second quadrant points, P and S₂ 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 S₁ 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 S₁ and the abscissas of the points S₂ 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, S₁ and S₂, 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.

Pre-calculus contents A