Machinery's Handbook, 31st Edition
TRIGONOMETRIC FUNCTIONS 105 Trigonometric Functions.—Like algebraic functions, trigonometric functions define a relationship between input values (angle measure, expressed in radians) and output values (the trigonometric ratios associated with the angles). Recalling function notation, y = f ( x ) (see ALGEBRA on page 24), the main trigonometric functions are: f ( x ) = sin x , f ( x ) = cos x , and f ( x ) = tan x . The trigonometric graphs are derived from the points on the unit cir- cle (circle of radius = 1) “translated” onto the ( x , y )-coordinate system. Because the radius is 1, cos θ = x/ 1 = x , and sin θ = y/ 1 = y. Thus, the rectangular coordinate ordered pair ( x , y ) on the unit circle corresponds to the trigonometric coordinates (cos θ , sin θ ). Possible confusion arises from the different roles of x and y in the two graphs. On the unit circle (Fig. 1), x and y are the coordinates of the points that correspond to the adjacent (cosine) and opposite (sine) sides of the right triangle with radius 1. But on the function graphs (Fig. 2a), x is the angle measure , the same as θ , and y is the value of the function being plotted. So, “sin x ” is the same as “sin θ .” Angle measure is marked in radians ( not degrees ). The graphs of y = sin x and y = cos x are seen in the figure on the same set of axes (Fig. 2a). y = tan x is graphed on its own set of axes (Fig. 2b). The domain of a trigonometric function is the set of angle measures for which the func- tion is defined. The domain of both y = sin x and y = cos x is the set of all real numbers, since it makes sense to substitute any angle—positive or negative—into these functions. The answer is always a real number. Negative angles indicate an angle measured clockwise from the horizontal, whereas a positive angle (the usual situation) is measured counter- clockwise. The domain of y = tan x , however, does not include the odd multiples of π /2, since tan x = sin x /cos x , and so, when cos x = 0, tan x is undefined. This happens at ± π/2, ± 3π/2, ± 5π/2, . . . As Fig. 2b shows, the tangent function approaches the dotted lines in- creasingly closer, never meeting them.
Fig. 1. The Unit Circle, which Gives the Sine and Cosine Relationship for All Angle Measures.
Fig. 2. (a) Graph of y = sin x and y = cos x ; (b) Graph of y = tan x . Note: For any real number x , sin x (for example) is defined to be sin( x radians). Equivalent degree measures are shown here for reference. The signs of the three main trigonometric functions (positive or negative) are shown in the diagram at the top of page 106. The names of the positive-signed functions of angles that lie in a particular quadrant are shown. All the functions are positive in the first, only sin x in the second, only tan x in the third, and only cos x is positive in the fourth. The mne- monic device “ASTC” for “All Students Take Calculus” is a simple way to remember the signs. csc x , sec x , and cot x have the same signs as their respective reciprocal functions.
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