/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 37

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

Short Answer

Expert verified
Question: Find all values of x for which the equation \(\tan x = 1\) is true. Answer: The general solution for the equation \(\tan x = 1\) is: $$x = 45掳 + 180掳k = \frac{\pi}{4} + \pi k$$ where k is an integer.

Step by step solution

01

Identify the basic angles where tangent equals 1

First, we should recognize that tangent function has a periodicity of 180 degrees (or 蟺 radians). Therefore, after finding the basic angles where tangent equals 1, we can add the period of the tangent function to find all other solutions. The unit circle gives us an intuitive way to find these basic angles. From the unit circle, we know that: $$\tan x = \frac{\sin x}{\cos x}$$ We can find the first angle where the tangent equals 1 by noting that it occurs when the sine and cosine functions are equal. This happens at: $$x = 45掳 = \frac{\pi}{4}$$
02

Use the periodicity of the tangent function to find all solutions

Now that we have the basic angle for which the tangent is equal to 1, we can use the periodicity of the tangent function to find all other solutions. Since the tangent function repeats every 180 degrees (or 蟺 radians), our general solution will be: $$x = 45掳 + 180掳k = \frac{\pi}{4} + \pi k$$ where k is an integer.
03

Final answer:

The general solution for the equation \(\tan x = 1\) is: $$x = 45掳 + 180掳k = \frac{\pi}{4} + \pi k$$ where k is an integer.

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