II.

Plane Trigonometry

The concept of the trigonometric angle is basic to the study of trigonometry. A trigonometric angle is generated by a rotating ray. The rays OA and OB (Fig. 1a, 1b, and 1c) are considered originally coincident at OA, which is called the initial side. The ray OB then rotates to a final position called the terminal side. An angle and its measure are considered positive if they are generated by counterclockwise rotation in the plane, and negative if they are generated by clockwise rotation. Two trigonometric angles are equal if they are congruent and if their rotations are in the same direction and of the same magnitude.

An angular unit of measure usually is defined as an angle with a vertex at the center of a circle and with sides that subtend, or cut off, a certain part of the circumference (Fig. 2).

If the subtended arc s (AB) is equal to one-fourth of the total circumference C, that is, s = ‚C, so that OA is perpendicular to OB, the angular unit is a right angle. If s = yC, so that the points A, O, and B are on a straight line, the angular unit is a straight angle. If s = 1/360C, the angular unit is one degree. If s = C, so that the subtended arc is equal to the radius of the circle, the angular unit is a radian. By equating the various values of C, it follows that

1 straight angle = 2 right angles = 180 degrees = p radians

Each degree is subdivided into 60 equal parts called minutes, and each minute is subdivided into 60 equal parts called seconds. For finer measurements, decimal parts of a second may be used. Radian measurements smaller than a radian are expressed in decimals. The symbol for degree is °; for minutes, ‘; and for seconds, '. For radian measures either the abbreviation rad or no symbol at all may be used. Thus

The angular unit radian is understood in the last entry. (The notation 42'.14 may be used instead of 42.14' to indicate decimal parts of seconds.)

By convention, a trigonometric angle is labeled with the Greek letter theta (θ). If the angle θ is given in radians, then the formula s = rθ may be used to find the length of the arc s; if θ is given in degrees, then

A.

Trigonometric Functions

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.

In Fig. 3, let P, with coordinates x and y, be any point other than the vertex on the terminal side of the angle θ, 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 (see Geometry).

The six commonly used trigonometric functions are defined as follows:

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 (θ + 2p) = sin θ. Similar statements hold for the five other functions. By definition, three of these functions are reciprocals of the three others, that is,

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 θ and cos θ range from -1 to +1; tan θ and cot θ are unlimited, assuming any real value; sec θ and csc θ 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 θ is one of the acute angles of a right triangle, the definitions of the trigonometric functions given above can be applied to θ 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 θ = y/r = a/c, and so on, as follows:

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

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 θ and cos θ 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.

B.

Trigonometric Identities

The following formulas, called identities, which show the relationships between the trigonometric functions, hold for all values of the angle θ, or of two angles, θ and φ, for which the functions involved are:

By repeated use of one or more of the formulas in group V, which are known as reduction formulas, sin θ and cos θ can be expressed for any value of θ, 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 θ, cot θ, sec θ, and csc θ may be found from the values of sin θ and cos θ. It is therefore sufficient to tabulate the values of sin θ and cos θ for values of θ 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 θ.

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.

C.

Inverse Functions

The statement y is the sine of θ, or y = sin θ is equivalent to the statement θ is an angle, the sine of which is equal to y, written symbolically as θ = arc sin y = sin^-1y. 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 θ, or θ = arc sin y, a given value of y will determine infinitely many values of θ. Thus, sin 30° = sin 150° = sin (30° + 360°) = sin (150° + 360°). . .= 1/2; therefore, if θ = arc sin 1/2, then θ = 30° + n360° or θ = 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

D.

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

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.