Trigonometric
functions are unitless values that vary with the size of an angle.
An angle placed in a rectangular coordinate plane is said to
be in standard position if its vertex coincides with the origin
and its initial side coincides with the positive x-axis.
Sine and
Cosecant
Sine and cosecant
are periodic trigonometric functions-that is, their values repeat
at regular intervals called periods. The period of sine and cosecant
is 360°, or 2p.
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Cosine and Secant
Cosine and secant
are periodic trigonometric functions-that is, their values repeat
at regular intervals called periods. The period of cosine and
secant is 360°, or 2p.
Tangent and Cotangent
Tangent and
cotangent are periodic trigonometric functions-that is, their
values repeat at regular intervals called periods. The period
of tangent and cotangent is 180°, or p.
Figure 5: Periodic Functions
The values
of periodic trigonometric functions repeat at regular intervals.
This graph shows that tangent and cotangent functions have periods
of 360° or 2p radians, while all other trigonometric functions
have periods of 180° or p radians
- In Fig. 3, let P,
with coordinates x and y, be any point other than the vertex
on the terminal side of the angle q, and r be the distance between
Pand the origin. Each of the coordinates x and y may be positive
or negative, depending on the quadrant in which the point P lies;
x may be zero, if P is on the y- axis, or y may be zero, if P
is on the x-axis. The distance r is necessarily positive and
is equal to », in accordance with the Pythagorean theorem.
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- The six commonly used
trigonometric functions are defined as follows:
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- Since x and y do not
change if 2p radians are added to the angle-that is, 360°
are added-it is clear that sin (q + 2p) = sin q. Similar
statements hold
for the five other functions. By definition, three of these functions
are reciprocals of the three others, that is,
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- If point P, in the
definition of the general trigonometric function, is on the y-axis,
x is 0; therefore, because division by zero is inadmissible in
mathematics, the tangent and secant of such angles as 90°,
270°, and -270° do not exist. If P is on the x-axis,
y is 0; in this case, the cotangent and cosecant of such angles
as 0°, 180°, and -180° do not exist. All angles have
sines and cosines, because r is never equal to 0.
- Since r is greater
than or equal to x or y, the values of sin q and cos q range
from -1 to +1; tan q and cot q are unlimited, assuming any real
value; sec q and csc q may be either equal to or greater than
1, or equal to or less than -1.
- It is readily shown
that the value of a trigonometric function of an angle does not
depend on the particular choice of point P, provided that it
is on the terminal side of the angle, because the ratios depend
only on the size of the angle, not on where the point P is located
on the side of the angle.
- If q is one of the
acute angles of a right triangle, the definitions of the trigonometric
functions given above can be applied to q as follows (Fig. 4).
Imagine the vertex A is placed at the intersection of the x-axis
and y-axis in Fig. 3, that AC extends along the positive x-axis,
and that B is the point P, so that AB = AP = r. Then sin q =
y/r = a/c, and so on, as follows:
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- The numerical values
of the trigonometric functions of a few angles can be readily
obtained; for example, either acute angle of an isosceles right
triangle is 45°, as shown in Fig. 4. Therefore, it follows
that
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- The numerical values
of the trigonometric functions of any angle can be determined
approximately by drawing the angle in standard position with
a ruler, compass, and protractor; by measuring x, y, and r; and
then by calculating the appropriate ratios. Actually, it is necessary
to calculate the values of sin q and cos q only for a few selected
angles, because the values for other angles and for the other
functions may be found by using one or more of the trigonometric
identities that are listed below.
- Trigonometric Identities
The following formulas,
called identities, which show the relationships between the trigonometric
functions, hold for all values of the angle q, or of two angles,
q and f, for which the functions involved are:
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- By repeated use of
one or more of the formulas in group V, which are known as reduction
formulas, sin q and cos q can be expressed for any value of q,
in terms of the sine and cosine of angles between 0° and
90°. By use of the formulas in groups I and II, the values
of tan q, cot q, sec q, and csc q may be found from the values
of sin q and cos q. It is therefore sufficient to tabulate the
values of sin q and cos q for values of q between 0° and
90°; in practice, to avoid tedious calculations, the values
of the other four functions also have been made available in
tabulations for the same range of q.
- The variation of the
values of the trigonometric functions for different angles may
be represented by graphs, as in Fig. 5. It is readily ascertained
from these curves that each of the trigonometric functions is
periodic, that is, the value of each is repeated at regular intervals
called periods. The period of all the functions, except the tangent
and the cotangent, is 360°, or 2 p radians. Tangent and cotangent
have a period of 180°, or p radians.
- Many other trigonometric
identities can be derived from the fundamental identities. All
are needed for the applications and further study of trigonometry.
- Inverse Functions
The statement y
is the sine of q, or y = sin q is equivalent to the statement
q is an angle, the sine of which is equal to y, written symbolically
as q = arc sin y = sin-1 y. The arc form is preferred. The inverse
functions, arc cos y, arc tan y, arc cot y, arc sec y, arc csc
y, are similarly defined. In the statement y = sin q, or q =
arc sin y, a given value of q will determine infinitely many
values of y. Thus, sin 30° = sin 150° = sin (30°
+ 360°) = sin (150° + 360°). . .= 1/2; therefore,
if q = arc sin 1/2, then q = 30° + n360° or q = 150°
+ n360°, in which n is any integer, positive, negative, or
zero. The value 30° is designated the basic or principal
value of arc sin 1/2. When used in this sense, the term arc generally
is written with a capital A. Although custom is not uniform,
the principal value of Arc sin y, Arc cos y, Arc tan y, Arc cot
y, Arc sec y, or Arc csc y commonly is defined to be the angle
between 0° and 90° if y is positive; and, if y is negative,
by the inequalities
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- The General
Triangle
Practical applications
of trigonometry often involve determining distances that cannot
be measured directly. Such a problem may be solved by making
the required distance one side of a triangle, measuring othersides
or angles of the triangle, and then applying the formulas below.
- If A, B, C are the
three angles of a triangle, and a, b, c the respective opposite
sides, it may be proved that
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- The cosine and tangent
laws can each be given two other forms by rotating the letters
a, b, c and A, B, C.
- These three relationships
can be used to solve any triangle, that is, the unknown sides
or angles can be found when one side and two angles, two sides
and the included angle, two sides and an angle opposite one of
them (usually there are two triangles in this case), or when
three sides are given.
- Spherical Trigonometry
Spherical trigonometry,
which is used principally in navigation and astronomy, is concerned
with spherical triangles, that is, figures that are arcs of great
circles on the surface of a sphere. The spherical triangle, like
the plane triangle, has six elements, the three sides a, b, c
and the angles A, B, C. But the three sides of the spherical
triangle are angular as well as linear magnitudes, being arcs
of great circles on the surface of a sphere and measured by the
angle subtended at the center. The triangle is completely determined
when any three of its six elements are given, since relations
exist between the various parts by means of which unknown elements
may be found.
- In the right-angled
or quadrantal triangle, however, as in the case of the right-angled
plane triangle, only two elements are needed to determine all
of the remaining parts. Thus, given c, A in the right-angled
triangle, ABC, with C = 90°, the remaining parts are given
by the formula as sin a = sin c sin A; tan b = tan c cos A; cot
B = cos c tan A. When any other two parts are given the corresponding
formulas may be obtained by Napier's rules concerning the relations
of the five circular parts, a, b, complement of c, complement
of A, complement of B. With respect to any particular part, the
remaining parts are classified as adjacent and opposite; the
sine of any part is equal to the product of the tangents of the
adjacent parts and also to the product of the cosines of the
opposite parts.
- In the case of oblique
triangles no simple rules have been found, but each case depends
on the appropriate formula. Thus in the oblique triangle ABC,
given a, b, and A, the formulas for the remaining parts are
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- In spherical trigonometry,
as well as in plane, three elements taken at random may not satisfy
the conditions for a triangle, or they may satisfy the conditions
for more than one. The treatment of certain cases in spherical
trigonometry is quite formidable, because every line intersects
every other line in two points and multiplies the cases to be
considered. The measurement of spherical polygons may be made
to depend upon that of the triangle. If, by drawing diagonals,
one can divide the polygons into triangles, each of which contains
three known or obtainable elements, then all the parts of the
polygon can be determined.
- Spherical trigonometry
is of great importance in the theory of stereographic projection
and in geodesy. It is also the basis of the chief calculations
of astronomy; for example, the solution of the so-called astronomical
triangle is involved in finding the latitude and longitude of
a place, the time of day, the position of a star, and various
other data.
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