Trigonometry and 3-Space
Evaluating Trigonometric Ratios for any Angle between 0 and 360
RELATING TRIGONOMETRIC RATIOS TO A POINT IN QUADRANT 1 OF THE CARTESIAN PLANE


Determine the value of r.

Label x, y, and r on your diagram.

Determine the primary trigonometric ratios for the
principal angle.
Determine the reciprocal trigonometric ratios for the principal angle.
RELATING TRIGONOMETRIC RATIOS TO A POINT IN QUADRANT 2 OF THE CARTESIAN PLANE

Label x, y and r on your diagram

Using the related acute angle, determine the primary
trigonometric ratios for the principal angle  .

Determine the reciprocal trigonometric ratios for the
principal angle.
RELATING TRIGONOMETRIC RATIOS TO A POINT IN QUADRANT 3 OF THE CARTESIAN PLANE

Label, x, y, and r on your diagram.

Using the related acute angle, determine the primary
trigonometric ratios for the principal angle.

Determine the reciprocal trigonometric ratios for the
principal angle.
Trigonometry and 3-Space
RELATING TRIGONOMETRIC RATIOS TO A POINT IN QUADRANT 4 OF THE CARTESIAN PLANE

Label x, y, and r.

Using the related acute angle, determine the primary
trigonometric ratios for the principal angle.

Determine the reciprocal trigonometric ratios for the
principal angle.
CONCLUSIONS:

Record the sign
( + or - ) for each of the primary
trigonometric ratios in each quadrant.

In which quadrant(s) is sine positive?

In which quadrant(s) is cosine positive?

In which quadrant(s) is tangent
positive?
For any principal angle greater than 90 , the
values of the primary trigonometric ratios are
either the same as, or the negatives of, the ratios
for the related acute angle. These relationships
are based on angles in standard position in the
Cartesian plane and depend on the quadrant in
which the terminal arm of the angle lies.
For any point P (x,y) in the Cartesian plane, the
trigonometric ratios for angles in standard position
can be expressed in terms of x, y and r.
y
x
y
sin  
cos  
tan  
r
r
x
Trigonometry and 3-Space
The principal angle and the related acute angle are the
same angle in quadrant 1.
If  is an acute angle in standard position, then
the terminal arm of the principal angle
(180   ) lies in quadrant 2.
If  is an acute angle in standard position,
then the terminal arm of the principal angle,
(180   ) lies in quadrant 3.
If  is an acute angle in standard position,
then the terminal arm of the principal angle, (360   )
lies in quadrant 4.