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Chapter 5: Problem 37

Verify each identity. $$\tan (2 \pi-x)=-\tan x$$

Short Answer

Expert verified
Upon applying the property of the negative tangent function and utilizing the periodicity of the tangent function, we can successfully verify that the given identity \( \tan(2\pi - x) = -\tan(x) \) is correct.

Step by step solution

01

Applying the property of tangent

We begin by recalling the property of the tangent function: \( \tan(-x) = -\tan(x) \). Let's apply that to our equation and replace \(\tan(2\pi - x)\) with \( -\tan(x-2\pi)\). In this step, we use the property of tangent for a difference of angles.
02

Applying the periodicity of tangent

We now apply the property of periodicity of the tangent function: \( \tan(x + n\pi) = \tan(x) \) for all integer \( n \). Replacing \(x-2\pi\) with \(x\), our expression becomes \( -\tan(x) \). We can verify that it matches our right-hand side, completing the verification of the identity.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 101 to answer each of the following. a. Is \(\sin \left(30^{\circ}+60^{\circ}\right),\) or \(\sin 90^{\circ},\) equal to \(\sin 30^{\circ}+\sin 60^{\circ} ?\) b. Is \(\sin \left(30^{\circ}+60^{\circ}\right),\) or \(\sin 90^{\circ},\) equal to \(\sin 30^{\circ} \cos 60^{\circ}+\cos 30^{\circ} \sin 60^{\circ} ?\)

Will help you prepare for the material covered in the next section. Give exact values for \(\cos 30^{\circ}, \sin 30^{\circ}, \cos 60^{\circ}, \sin 60^{\circ}, \cos 90^{\circ}\) and \(\sin 90^{\circ}\)

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \sin ^{2} x=2-3 \sin x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning.After using an identity to determine the exact value of \(\sin 105^{\circ}, 1\) verified the result with a calculator.

Determine the amplitude and period of \(y=3 \cos 2 \pi x\) Then graph the function for \(-4 \leq x \leq 4\) (Section 4.5, Example 5)
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