Classify the reciprocal trigonometric functions as odd or even, andthen write the corresponding equation.b) Use transformations to explain why each equation is true.
Question
Classify the reciprocal trigonometric functions as odd or even, andthen write the corresponding equation.b) Use transformations to explain why each equation is true.
Solution
The reciprocal trigonometric functions are cosecant (csc), secant (sec), and cotangent (cot).
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Cosecant (csc): The cosecant function is an odd function because it satisfies the property csc(-x) = -csc(x). This means that the function is symmetric with respect to the origin.
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Secant (sec): The secant function is an even function because it satisfies the property sec(-x) = sec(x). This means that the function is symmetric with respect to the y-axis.
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Cotangent (cot): The cotangent function is an odd function because it satisfies the property cot(-x) = -cot(x). This means that the function is symmetric with respect to the origin.
The transformations that explain why each equation is true are as follows:
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For the cosecant function, a reflection in the y-axis followed by a reflection in the x-axis gives the original function, which is the definition of an odd function.
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For the secant function, a reflection in the y-axis gives the original function, which is the definition of an even function.
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For the cotangent function, a reflection in the y-axis followed by a reflection in the x-axis gives the original function, which is the definition of an odd function.
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