Principles of trigonometry animated

Trigonometry
History of Trigonometry
Principles of Trigonometry

Principles of Trigonometry
Trigonometric Function
Plane Trigonometry
Spherical Trigonometry
Analytic Trigonometry
Coordinates and transformation of
coordinates

Trigonometric Functions
• Trigonometric Function of an
Angle
• Tables of Natural Function
PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY

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.
•

• 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.

• 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.

• 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.

• 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).

• 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.

• 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.

• 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.
•

• 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.

Trigonometric functions of an
angle
•
PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
•

angle
•
•
•

angle
•

angle
•
•

angle
•
•
•

Tables of Natural Functions
•
•

Tables of Natural Functions
•
′
•

Plane Trigonometry
•
•
•

Plane Trigonometry
•
•
• 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

Plane Trigonometry
•

Plane Trigonometry
•
•

•
•

•
•
α β γ

•
α β γ

•

•
•
•

•
•

Coordinates and
Transformation of Coordinates
• Polar Coordinates
• Transformation of
Coordinates

Polar Coordinates
•
θ
θ
•
PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF
COORDINATE
•
θ

Polar Coordinates
• θ θ
θ
θ
•
θ θ θ
COORDINATE

Polar Coordinates
•
θ θ
• θ
COORDINATE

Polar Coordinates
•
θ
ϕ
θ θ ϕ
θ ϕ
• θ ϕ
COORDINATE

Transformation of
Coordinates
•
•
COORDINATE

Transformation of
Coordinates
•
θ
θ
COORDINATE

Transformation of
Coordinates
•
′ ′
′ ′
•
COORDINATE

Transformation of
Coordinates
•
′ ′
′ ′
COORDINATE

Transformation of
Coordinates
•
ϕ
′ ′
•
′ ϕ ′ ϕ
′ ϕ ′ ϕ
COORDINATE
•
ϕ

Transformation of
Coordinates
•
ϕ
′ ′
•
COORDINATE

Principles of trigonometry animated

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Principles of trigonometry animated