/*! 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 47 Find all of the angles which sat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 47

Find all of the angles which satisfy the equation. $$ \tan (\theta)=0 $$

Short Answer

Expert verified
The angles satisfying \( \tan(\theta) = 0 \) are \( \theta = n\pi \), where \( n \) is an integer.

Step by step solution

01

Understand Tangent Zero Points

The tangent function, \( \tan(\theta) \), equals zero when the angle \( \theta \) is at the origin and at its periodic intervals on the unit circle. Typically, these points occur at specific angles.
02

Identify Primary Angle

In trigonometry, \( \tan(\theta) = 0 \) happens at \( \theta = 0 \) since \( \tan(\theta) \) is the sine over cosine, \( \frac{\sin(\theta)}{\cos(\theta)} \), and \( \sin(0) = 0 \).
03

Consider the Period of Tangent

The tangent function is periodic with a period of \( \pi \). This means \( \tan(\theta) = \tan(\theta + k\pi) \) for any integer \( k \). This gives us the general solution for \( \theta \).
04

Write the General Solution

Using the periodic nature of the tangent function, the angles that satisfy \( \tan(\theta) = 0 \) are \( \theta = n\pi \) where \( n \) is any integer.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Tangent Function
The tangent function, often denoted as \( \tan(\theta) \), is a fundamental trigonometric function. It's defined as the ratio of the sine and cosine of an angle: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This means that when the sine of an angle is zero, the tangent of the angle is also zero, provided the cosine is not zero.

Tangent has some unique characteristics:
  • Range: The range of the tangent function is all real numbers because it extends from negative to positive infinity as \( \theta \) approaches the vertical asymptotes.
  • Asymptotes: Tangent has vertical asymptotes where the cosine is zero, which occur at \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
  • Zero Points: The function returns to zero at integer multiples of \( \pi \) (e.g., \( 0, \pi, 2\pi \)). These are key to solving equations where \( \tan(\theta) = 0 \).
General Solution
In trigonometry, a general solution gives all possible values of an angle that satisfy a trigonometric equation. For the equation \( \tan(\theta) = 0 \), the general solution is derived from understanding where the tangent function equals zero. As we've discussed, the tangent function returns a zero value at every multiple of \( \pi \).

This gives the general solution:
  • \( \theta = n\pi \), where \( n \) is an integer, indicating that all angles which are whole number multiples of \( \pi \) will satisfy \( \tan(\theta) = 0 \).
This method essentially accounts for all angles across the infinite span of the unit circle, capturing the periodic nature of the tangent function.
Periodic Functions
Periodic functions are those that repeat their values in regular intervals. The tangent function is one such function, characterized by its interval, known as the period, which defines the length over which its values repeat before starting again.

For \( \tan(\theta) \), the period is \( \pi \):
  • This means that every \( \pi \) interval, from 0 to \( \pi \), \( \pi \) to \( 2\pi \), etc., the pattern of the tangent's values repeats.
  • Periodic functions like tangent allow mathematicians to predict and confirm solutions across an infinite set of cycles without having to trace each value manually.
The concept of periodicity is central to solving trigonometric equations since it provides a way to express infinite solutions concisely. Hence, for \( \tan(\theta) = 0 \), using \( n\pi \), we can succinctly capture every solution due to this periodic behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{\pi}{6}\right)\right) $$

In Exercises \(165-184\), rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. If \(\sec (\theta)=\frac{x}{4}\) for \(0<\theta<\frac{\pi}{2}\), find an expression for \(4 \tan (\theta)-4 \theta\) in terms of \(x .\)

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \cos (\operatorname{arccot}(3)) $$

In Exercises \(119-130\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{11 \pi}{6}\right)\right) $$

If \(\sec (\theta)=\frac{x}{4}\) for \(0<\theta<\frac{\pi}{2}\), find an expression for \(\ln |\sec (\theta)+\tan (\theta)|\) in terms of \(x\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.