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Chapter 7: Problem 6

Solve, finding all solutions. Express the solutions in both radians and degrees. $$\tan x=-1$$

Short Answer

Expert verified
In radians: \(\frac{3\pi}{4} + k\pi\). In degrees: 135\^{\circ} + 180\^{\circ} \cdot k

Step by step solution

01

Understand the Problem

The equation to solve is \(\tan x = -1\). The goal is to find the values of \(x\) where the tangent function equals \-1\. Solutions should be expressed in both radians and degrees.
02

Find the General Solutions

The tangent function is periodic with a period of \( \pi \). The general solution for \( \tan{x} = -1 \) is given by \( x = \frac{3\pi}{4} + k\pi \), where \( k \) is any integer. This includes all angles where the tangent is equal to \-1\.
03

Express the Principal Solutions in Radians

The principal solutions within one period \( [0, \pi) \) are: \(\frac{3\pi}{4}\) radians and \(\frac{7\pi}{4}\) radians. To find more solutions, use the general solution formula.
04

Convert the Principal Solutions to Degrees

Convert \( \frac{3\pi}{4} \) radians to degrees by multiplying by \( \frac{180}{\pi}\): \( \frac{3\pi}{4} \cdot \frac{180}{\pi} = 135^{\circ}\). Similarly, convert \(\frac{7\pi}{4}\) radians: \(\frac{7\pi}{4} \cdot \frac{180}{\pi} = 315^{\circ}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

tangent function
The tangent function, denoted as \tan\ x\, is one of the fundamental trigonometric functions. It is defined as the ratio of the sine function to the cosine function \( \tan x = \frac{\sin x}{\cos x} \).

When solving the equation \( \tan x = -1 \), we seek the angles \( x \) where this condition holds true.

Note that the tangent function has some unique properties:
  • It is undefined where \( \cos x = 0 \).
  • Its values range between \(-\infty\) and \(\infty\).
Realizing these properties helps in accurately resolving trigonometric equations involving the tangent function.
radians to degrees conversion
To solve trigonometric equations completely, it's essential to express solutions in both radians and degrees.

Radians and degrees are two units measuring angles. The key conversion formula is: \[ 1 \text{ radian} \approx 57.2958^{\circ} \] Or more commonly, \[ \text{Degrees} = \text{Radians} \times \left(\frac{180}{\pi}\right)\]

For example, for the general solutions in our problem, to convert \( \frac{3\pi}{4} \) radians to degrees:
  • Multiply by \( \frac{180}{\pi} \): \[ \frac{3\pi}{4} \times \frac{180}{\pi} = 135^{\circ} \]
Likewise, converting \( \frac{7\pi}{4} \) radians to degrees:
  • Multiply by \( \frac{180}{\pi} \): \[ \frac{7\pi}{4} \times \frac{180}{\pi} = 315^{\circ} \]
These conversions are crucial when providing complete solutions in both formats as requested in our problem.
periodic functions
Understanding periodic functions is vital when solving trigonometric equations involving the tangent function.

A function is periodic if it repeats its values in regular intervals or periods. The period of the tangent function is \( \pi \), meaning:
  • \( \tan(x + \pi) = \tan x \)
This property is what allows us to generalize solutions. For \( \tan x = -1 \), if \( x = \frac{3\pi}{4} \) solves the equation, then \( \frac{3\pi}{4} + k\pi \) (where \( k \) is any integer) also solves it.

Recognizing these periodic properties helps us find all possible solutions efficiently. Often, principal solutions (within one period) are identified first, followed by extending using the periodicity property.

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Most popular questions from this chapter

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