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Chapter 1: Problem 37

Solve the following equations. $$\tan x=1$$

Short Answer

Expert verified
Answer: The solutions of the given equation are $$x = \frac{\pi}{4} + n\pi$$, where n is an integer.

Step by step solution

01

Identify Periodicity and Symmetry

The tangent function has a period of $$\pi$$, meaning that $$\tan(x + k\pi) = \tan(x)$$ for any integer k. Additionally, it is an odd function, i.e., $$\tan(-x) = - \tan(x)$$.
02

Solve the Equation for the First Quadrant

In the first quadrant (between 0 and $$\frac{\pi}{2}$$), we know that the tangent function is positive. Given $$\tan{x} = 1$$, we can find the value of x by using the fact that the tangent function is equal to unity at $$x = \frac{\pi}{4}$$, therefore, $$x = \frac{\pi}{4}$$.
03

Find General Solution

Since tangent has a period of $$\pi$$, the general solution can be expressed as $$x = \frac{\pi}{4} + n\pi$$, where n is an integer. x-axis in the form = 1, we can find the value of x by knowing the source of the function Similarly, considering the symmetry property (odd function), the equation $$\tan(-x) = -\tan(x)$$ implies that the solution in the opposite direction (negative x) is of equal magnitude but opposite sign. Therefore, the solutions of the given equation are $$x = \frac{\pi}{4} + n\pi$$, where n is an integer.

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Most popular questions from this chapter

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\sin (a+b)=\sin a+\sin b\) b. The equation \(\cos \theta=2\) has multiple real solutions. c. The equation \(\sin \theta=\frac{1}{2}\) has exactly one solution. d. The function \(\sin (\pi x / 12)\) has a period of 12 e. Of the six basic trigonometric functions, only tangent and cotangent have a range of \((-\infty, \infty)\) f. \(\frac{\sin ^{-1} x}{\cos ^{-1} x}=\tan ^{-1} x\) g. \(\cos ^{-1}(\cos (15 \pi / 16))=15 \pi / 16\) h. \(\sin ^{-1} x=1 / \sin x\)

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Assume that \(b > 0\) and \(b \neq 1\). Show that \(\log _{1 / b} x=-\log _{b} x\).

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