Using Even and Odd Trigonometric Functions
To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function (f(x)=x^2), shown in Figure (PageIndex{5}). The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: ((4)^2=(−4)^2,(−5)^2=(5)^2), and so on. So (f(x)=x^2) is an even function, a function such that two inputs that are opposites have the same output. That means (f(−x)=f(x)).
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Now consider the function (f(x)=x^3), shown in Figure (PageIndex{6}). The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So (f(x)=x^3) is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means (f(−x)=−f(x)).
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We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure (PageIndex{7}). The sine of the positive angle is (y). The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table (PageIndex{2}).
Table (PageIndex{2}) (begin{align} sin t &=y sin (−t) &=−y sin t &≠sin(−t) end{align}) ( begin{align} cos t &=x cos (−t)=x cos t &= cos (−t) end{align}) (begin{align} tan (t) &= frac{y}{x} tan (−t) &=−frac{y}{x} tan t &≠ tan (−t) end{align}) (begin{align} sec t &= frac{1}{x} sec (−t) &= frac{1}{x} sec t &= sec (−t) end{align}) ( begin{align} csc t &= frac{1}{y} csc (−t) &= frac{1}{−y} csc t &≠ csc (−t) end{align}) ( begin{align} cot t &= frac{x}{y} cot (−t) &= frac{x}{−y} cot t & ≠ cot (−t) end{align})
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Category: WHY
