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Chapter 4: Problem 56

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
By comparing the values for positive and negative angles in the unit circle, it can be verified that cosine and secant functions are even, while sine, cosecant, tangent, and cotangent functions are odd.

Step by step solution

01

Verify even functions

For any angle \(\theta\) in the unit circle, \(\cos(\theta) = \cos(-\theta)\) and \(\sec(\theta) = \sec(-\theta)\), which satisfies the definition of an even function. As an example, \(\cos(30^{\circ}) = \cos(-30^{\circ}) = \sqrt{3}/2\) and \(\sec(30^{\circ}) = \sec(-30^{\circ}) = 2/\sqrt{3}\).
02

Verify odd functions

For any angle \(\theta\) in the unit circle, \(\sin(\theta) = -\sin(-\theta)\), \(\csc(\theta) = -\csc(-\theta)\), \(\tan(\theta) = -\tan(-\theta)\), and \(\cot(\theta) = -\cot(-\theta)\), which satisfies the definition of an odd function. As an example, \(\sin(30^{\circ}) = -\sin(-30^{\circ}) = 1/2\) and \(\tan(30^{\circ}) = -\tan(-30^{\circ}) = \sqrt{3}/3\).

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