4. Trigonometric functions
• A somewhat more general concept of
angle is required for trigonometry than
for geometry.
• An angle A with vertex at V, the initial
side of which is VP and the terminal
side of which is VQ, is indicated in the
figure by the solid circular arc.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
•
5. Trigonometric functions
• This angle is generated by the continuous
counterclockwise rotation of a line
segment about the point V from the
position VP to the position VQ.
• A second angle A′ with the same initial
and terminal sides, indicated in the
figure by the broken circular arc, is
generated by the clockwise rotation of
the line segment from the position VP to
the position VQ.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
6. Trigonometric functions
• Angles are considered positive when
generated by counterclockwise
rotations, negative when generated by
clockwise rotations.
• The positive angle A and the negative
angle A′ in the figure are generated by
less than one complete rotation of the
line segment about the point V.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
7. Trigonometric functions
• All other positive and negative angles
with the same initial and terminal sides
are obtained by rotating the line
segment one or more complete turns
before coming to rest at VQ.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
8. Trigonometric functions
• Numerical values can be assigned to
angles by selecting a unit of measure.
• The most common units are the degree
and the radian.
• There are 360° in a complete
revolution, with each degree further
divided into 60′ (minutes) and each
minute divided into 60″ (seconds).
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
9. Trigonometric functions
• In theoretical work, the radian is the
most convenient unit.
• It is the angle at the centre of a circle
that intercepts an arc equal in length
to the radius; simply put, there are
2π radians in one complete revolution.
• From these definitions, it follows that
1° = π/180 radians.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
10. Trigonometric functions
• Equal angles are angles with the same
measure; i.e., they have the same sign
and the same number of degrees.
• Any angle −A has the same number of
degrees as A but is of opposite sign.
• Its measure, therefore, is the negative
of the measure of A.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
11. Trigonometric functions
• If two angles, A and B, have the initial
sides VP and VQ and the terminal sides
VQ and VR, respectively, then the angle
A + B has the initial and terminal sides
VP and VR (see the figure).
• The angle A + B is called the sum of the
angles A and B, and its relation to A and
B when A is positive and B is positive or
negative is illustrated in the figure.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
•
12. Trigonometric functions
• The sum A + B is the angle the measure of
which is the algebraic sum of the measures
of A and B. The difference A − B is the sum
of A and −B.
• Thus, all angles coterminal with angle A (i.e.,
with the same initial and terminal sides as
angle A) are given by A ± 360n, in which
360n is an angle of n complete revolutions.
• The angles (180 − A) and (90 − A) are the
supplement and complement of angle A,
respectively.
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
28. Plane Trigonometry
•
•
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
• In addition to the
angles (A, B, C) and
sides (a, b, c), one of
the three heights of
the triangle (h) is
included by drawing
the line segment from
one of the triangle's
vertices (in this case
C) that is
Standard lettering
of a triangle