4.1 The Trigonometric Functions Sine, Cosine, and Tangent

When decomposing vectors into components or multiplying vectors by vectors, trigonometric functions come into play. This chapter serves to remind the reader of the definition of these functions. Those who remember the meaning of these functions sufficiently well from their schooldays can go straight to the next section.

Sides a and b of a right-angled triangle enclose the 90° angle ( Fig. 4.1 ); side c opposite this angle is termed the hypotenuse. In relation to the angle α between b and c, side a is termed opposite and side b adjacent. The trigonometric functions sine, cosine, and tangent describe the quotients of the lengths of the sides of right-angled triangles dependent on the angle α Using quotients of side lengths makes sense because all right-angled triangles with an angle α between the hypotenuse and the adjacent side are geometrically similar ( Fig. 4.1a ). In other words, for all right-angled triangles with an angle α between the hypotenuse and the adjacent side, the value of the quotient a/b is identical, irrespective of the actual size of the triangles.

The sine of an angle α is defined as

The cosine of an angle α is defined as

The tangent of an angle α is defined as

Of course the definitions hold for the angle β as well. The sine of angle β in Fig. 4.1b is defined as

Analogous definitions hold for the cosine and the tangent of angle β.

The trigonometric functions are frequently employed because many geometrical problems can be reduced to problems with right-angled triangles. If some elements (side lengths, angles) of a right-angled triangle are known, the remaining elements can be calculated trigonometrically.

Example 1: The length of the hypotenuse c and the angle α between the hypotenuse and one side are known. The trigonometric functions allow the lengths of sides a and b to be calculated. The definitions given above yield

Example 2: Angle α and the length of side b are known. The length of side a is calculated by

and the length of side c from

Numerical values of sine, cosine, and tangent can be found in mathematical tables or by means of a pocket calculator. In mathematical formulae the inverse functions of sine, cosine, and tangent are termed asin, acos, and atan. The inverse functions are the angles which correspond to quotients of side lengths of right-angled triangles: asin is the angle α that belongs to a given quotient of the lengths of sides a and c; by analogy, the same applies to acos and atan. Given the length of two sides of a right-angled triangle, the angle between the two sides can be determined using inverse trigonometric functions.

Example 3: The lengths of sides a and b are known. The angle α that belongs to the quotient a/b—that is, the angle between sides c and b—is calculated by

Values of the inverse trigonometric functions are also laid down in mathematical tables or stored in pocket calculators. On the buttons of such calculators, the inverse functions asin, acos, and atan are often designated as sin^-1, cos^-1, and tan^-1. (This is an awkward designation, because the symbol “^-1” is normally used to designate “1.0 divided by ….” The expression sin^-1α, however, does not designate 1.0/sinα but the inverse function of the sine, that is, the angle α belonging to a given value of the sine.)

As quotients of lengths of the sides of right-angled triangles, the trigonometric functions are defined only for angles up to 90°. In this domain sine, cosine, and tangent have (as quotients of lengths) positive values. It has proved advantageous, however, to extend the definition of trigonometric functions up to an angle of 360°. The generalized definition relates to quotients of lengths at the unit circle ( Fig. 4.2 ). The unit circle is a circle whose radius (distance OC) is equal to 1. The magnitude of the sine of an angle is equal to distance BC, the magnitude of the cosine is equal to distance OB, and the magnitude of the tangent is equal to distance AD. The functions, however, bear different positive or negative signs depending on the quadrant in which OC is located. Quadrants I to IV are parts of the plane as partitioned by a rectangular coordinate system. The signs for trigonometric functions are listed in Table 4.1 .

In the example shown in Fig. 4.2 (α = 125°), the sine of the angle α = 125° is positive; the cosine and the tangent are negative. The reader might like to confirm that, for angles up to 90°, defining trigonometric functions based on quotients of sides of right-angled triangles is identical to basing them on lengths at the unit circle.

Angles are quoted in different units, in degrees or radians. Expressed in degrees, a full rotation corresponds to 360°; expressed in radians, a full rotation corresponds to 2 · π = 6.2831 radians (correct to four decimal places). It follows that 1 radian corresponds to 57.2958°. The degree measure is preferred in practice; computerized calculation programs usually use the radian.

4.2 Representation of Vectors

In graphical illustrations, a vector is represented by an arrow ( Fig. 4.3 ). The length of the arrow indicates the magnitude of the vector. The magnitude of a vector is a positive quantity (there are no arrows with a negative length). In order to infer the magnitude of a vector from a graphical representation, the scale factor of the representation must be known. If, for example, the arrow represents a force, it must be known how many centimeters of its length correspond to 1 N. The direction of the vector is given by the direction of the arrow with its arrowhead. In a plane, the direction of a vector can be given by the angle α between the arrow and the x-axis of a rectangular xy-coordinate system and in addition by its arrowhead ( Fig. 4.4 ). In a plane, a vector is fully described by its magnitude and its direction. In three-dimensional space, an additional angle has to be given; this is usually the angle between the vector and the z-axis of a rectangular xyz-coordinate system.

Alternatively, a vector can be represented by its components in relation to an xyz-coordinate system. A vector F is then given by the vector sum of its components

In this formula e _x, e _y, and e _z designate unit vectors. Unit vectors are vectors of length 1 in the direction of the coordinate axes. F_x, F_y, and F_z are numbers which may be positive or negative. The x-component F_x · e _x of the vector F is a vector in the direction of the x-axis ( Fig. 4.4 ). The length of this vector equals the magnitude (the positive value) of F_x; the direction of this vector (the direction in which the arrow points) is given by the sign of F_x. For example, 5.0 · e _x designates a vector of length 5.0 pointing in the positive x-direction; −2.0 · e _x designates a vector of length 2.0 pointing in the negative x-direction. The same applies analogously to the other components of F.

In commonly used symbolic notation the component representation of a vector F is given by

In this notation, the unit vectors do not appear explicitly. The brackets indicate that the components of the vector F are calculated by multiplying the numbers F_x, F_y, and F_z by the unit vectors. If only vectors in the xy-plane are being dealt with, the z-component is always equal to zero. In this case, the third component can be omitted and the vector may be represented just by the two numbers F_x and F_y.

The numbers F_x, F_y, and F_z are known as the coordinates of the vector F. It has become usual to refer to F_x, F_y, and F_z as components of the vector F as well. Strictly speaking, this is not correct, because the components of a vector are vectors and not numbers. However, occasionally this somewhat loose use of the term component helps to prevent confusion when the coordinates of a point relative to a coordinate system are also being referred to.

Fig. 4.5 and Eqs. 4.11 and 4.12 illustrate the symbolic representation of vectors using the example of the two vectors F ₁ and F ₂ in the xy-plane. The axes of the rectangular coordinate system are scaled. The components of the vectors are the projections of the vectors on the axes; the signs of the coordinates give the direction of the components (the z-components in this example equal to zero).

Of course, this representation is valid for all vectors, not only for forces. A distance vector L ₁ in the xy-plane, pointing from the point (x = +1, y = +2) to the point (x = +6, y = +5) is represented by

The coordinates of a vector can be calculated from its magnitude and its direction. In the twodimensional case ( Fig. 4.4 ) it holds that

Conversely, the angle α can be determined when the coordinates F_x and F_y are known

The relation between the coordinates and the magnitude of a vector is given by

formula derives from ThisPythagoras’ theorem. The components F _x and F _y, together with the vector F, form a right-angled triangle. In a right-angled triangle

It must be pointed out that the location of the origin of a vector (the starting point of the arrow) is not specified by the vector’s magnitude and direction. The three vectors depicted in Fig. 4.6 have the same magnitude (length) and the same direction; they represent the identical vector. It follows that, when drawn, a vector may be shifted parallel to, or along, its direction. Since magnitude and direction remain unchanged by this procedure, it remains the identical vector. On the other hand, a mechanical state may well be altered when a vector is shifted. We expect different effects when the identical force vector is applied at different points of a body. The force might induce a linear translation and/or a rotation. To describe the effect of a force on a body it is not sufficient merely to communicate the force vector (magnitude and direction); the point of force application must be given as well.