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The Humongous Book of Trigonometry Problems by W. Michael Kelley

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Negative Identities

What happens when you plug in –x?

7.13    Evaluate each of the six trigonometric functions for –x to generate the negative trigonometric identities.

Consider the graphs of the trigonometric functions that you generate in Chapters 5 and 6. Any graph that is symmetric about the y-axis—in other words, the graph is a reflection of itself across the y-axis, as though that vertical line were a mirror—is described as an “even function.” If f(x) is an even function, then f(–x) = f(x). You get the same output when a real number and its opposite are substituted into the function.

Of the six trigonometric functions, only two have y-symmetric graphs and are, therefore, even functions: cos x and its reciprocal sec x. Therefore, cos (– ...

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