Which Trigonometric Function Is An Odd Function And Why - T-Tees - Your Answers Await, Life's Questions Unraveled

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.

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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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Figure (PageIndex{5}): The function (f(x)=x^2) is an even function.

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)).

Figure (PageIndex{6}): The function (f(x)=x^3) is an odd function.

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}).

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Figure (PageIndex{7})

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