121616949-math.79 - 4.1 Trigonometric Functions 65 to any...

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4.1 Trigonometric Functions 65 to any angles, as indicated in this figure: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x A (cos x, sin x ) The angle x subtends the heavy arc in the figure, that is, x = 7 π/ 6. Both coordinates of point A in this figure are negative, so the sine and cosine of 7 π/ 6 are both negative. The remaining trigonometric functions can be most easily defined in terms of the sine and cosine, as usual: tan x = sin x cos x cot x = cos x sin x sec x = 1 cos x csc x = 1 sin x and they can also be defined as the corresponding ratios of coordinates. Although the trigonometric functions are defined in terms of the unit circle, the unit circle diagram is not what we normally consider the graph of a trigonometric function. (The unit circle is the graph of, well, the circle.) We can easily get a qualitatively correct idea of the graphs of the trigonometric functions from the unit circle diagram. Consider the sine function, y = sin x . As x increases from 0 in the unit circle diagram, the second coordinate of the point A goes from 0 to a maximum of 1, then back to 0, then to a minimum of - 1, then back to 0, and then it obviously repeats itself. So the graph of y = sin x must look something like this: - 1 1 π/ 2 π 3 π/ 2 2 π - π/ 2 - π - 3 π/ 2 - 2 π .
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