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Chapter 4: Problem 66

Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd.

Short Answer

Expert verified
Verified that cosine and secant functions are even as \(\cos(x) = \cos(-x)\) and \(sec(x) = sec(-x)\) respectively. Verified that sine, cosecant, tangent and cotangent functions are odd as \(\sin(x) = -\sin(-x)\), \(csc(x) = -csc(-x)\), \(\tan(x) = -\tan(-x)\), and \(\cot(x) = -\cot(-x)\) respectively.

Step by step solution

01

Analyze Cosine Function (Even)

The cosine function of an angle in the unit circle is defined to be the x-coordinate of the point where the terminal side of that angle intersects the unit circle. If you choose a point on the unit circle associated with an angle \(x\), and another point associated with the angle \(-x\), these two will be reflected across the x-axis, hence they will have the same x-coordinate. Therefore, \(\cos(x) = \cos(-x)\) proving cosine is an even function.
02

Analyze Secant Function (Even)

The secant function is defined as the reciprocal of the cosine function, \(sec(x) = 1/\cos(x)\). Since cosine is an even function, its reciprocal is also an even function. Therefore, \(sec(x) = sec(-x)\) proving secant is an even function.
03

Analyze Sine (Odd)

The sine function of an angle in the unit circle is defined to be the y-coordinate of the point where the terminal side of that angle intersects the unit circle. If you consider an angle \(x\), and \(-x\), they will be reflections across the x-axis and therefore will have y-coordinates that are negatives of each other. Hence, \(\sin(x) = -\sin(-x)\) proving sine is an odd function.
04

Analyze Cosecant Function (Odd)

The cosecant function is defined as the reciprocal of the sine function, \(csc(x) = 1/\sin(x)\). Since sine is an odd function, so is its reciprocal. Therefore, \(csc(x) = -csc(-x)\) proving cosecant is an odd function.
05

Analyze Tangent Function (Odd)

The tangent function is defined as the ratio of the sine function to the cosine function. Since sine is odd and cosine is even, the ratio is an odd function, i.e. \(\tan(x) = -\tan(-x)\) proving tangent is an odd function.
06

Analyze Cotangent Function (Odd)

The cotangent function is the reciprocal of the tangent function. Since tangent is an odd function, its reciprocal is odd as well. Therefore, \(\cot(x) = -\cot(-x)\) proving cotangent is an odd function.

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