/*! 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 12 Solve the given equation. $$\t... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 12

Solve the given equation. $$\tan \theta-3 \cot \theta=0$$

Short Answer

Expert verified
\( \theta = \frac{\pi}{3} + n\pi \) or \( \theta = \frac{2\pi}{3} + n\pi \).

Step by step solution

01

Understand the Equation

The equation given is \( \tan \theta - 3 \cot \theta = 0 \). This implies that the tangent function of an angle minus three times the cotangent function of the same angle equals zero.
02

Rewrite Cotangent in Terms of Tangent

We know the cotangent function is the reciprocal of the tangent function, so \( \cot \theta = \frac{1}{\tan \theta} \). We substitute this into the equation to get \( \tan \theta - 3 \left(\frac{1}{\tan \theta}\right) = 0 \).
03

Eliminate the Fraction

Multiply the entire equation by \( \tan \theta \) to eliminate the fraction: \( \tan^2 \theta - 3 = 0 \).
04

Solve for \( \tan^2 \theta \)

Rearrange the equation to solve for \( \tan^2 \theta \): \( \tan^2 \theta = 3 \).
05

Solve for \( \tan \theta \)

Take the square root of both sides to solve for \( \tan \theta \): \( \tan \theta = \pm \sqrt{3} \).
06

Find General Solutions for \( \theta \)

From the values of \( \tan \theta \), we have two cases: 1) \( \tan \theta = \sqrt{3} \), which gives \( \theta = \frac{\pi}{3} + n\pi \), and 2) \( \tan \theta = -\sqrt{3} \), which gives \( \theta = \frac{2\pi}{3} + n\pi \), where \( n \) is an integer.

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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, often abbreviated as \( \tan \theta \), is a fundamental trigonometric function that relates an angle \( \theta \) of a right-angled triangle to the ratio of the opposite side to the adjacent side. In terms of the sine and cosine functions, it can be expressed as:
  • \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
The tangent function has some interesting properties, such as:
  • Periodicity: \( \tan \theta \) is periodic with a period of \( \pi \). This means the function repeats its values after every \( \pi \) radians.
  • Asymptotic behavior: \( \tan \theta \) has vertical asymptotes at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. At these points, \( \cos \theta = 0 \), making \( \tan \theta \) undefined.
Understanding these properties provides insight into solving equations involving \( \tan \theta \), like the one given in the original problem.
The solution involves simplifying and solving equations to find when \( \tan \theta \) equals specific values \( \sqrt{3} \) or \(-\sqrt{3} \).
Cotangent Function
The cotangent function, denoted as \( \cot \theta \), is another trigonometric function and is essentially the reciprocal of the tangent function. It can be expressed as:
  • \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)
This means that when \( \tan \theta = 0 \), \( \cot \theta \) becomes undefined due to division by zero. Conversely, when \( \sin \theta = 0 \), \( \cot \theta \) is undefined due to its definition involving division by the sine function.
The cotangent function has its own period and symmetry:
  • Periodicity: \( \cot \theta \) has a period of \( \pi \), much like \( \tan \theta \).
  • Symmetry: Like \( \tan \theta \), \( \cot \theta \) is an odd function, satisfying \( \cot(-\theta) = -\cot(\theta) \).
In solving trigonometric equations, understanding the relation \( \cot \theta = \frac{1}{\tan \theta} \) is crucial to rewrite and simplify expressions, as shown in the given solution.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable where the functions are defined. These identities are powerful tools for simplifying and solving trigonometric equations.
Two primary trigonometric identities relevant to the original exercise are:
  • Pythagorean identities: \( \tan^2 \theta + 1 = \sec^2 \theta \). These identities help in expressing or converting functions, especially when dealing with squared terms.
  • Reciprocal identities: \( \cot \theta = \frac{1}{\tan \theta} \) and \( \tan \theta = \frac{1}{\cot \theta} \). These are useful when changing one function into another, as in the step of replacing \( \cot \theta \) with its reciprocal.
Understanding these identities allows for the transformation and manipulation of trigonometric equations, making them easier to solve. Identifying these transformations in the problem helps solve for \( \theta \) more efficiently.

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Most popular questions from this chapter

Addition Formula for Tangent Use the Addition Formulas for cosine and sine to prove the Addition Formula for Tangent. \([\)Hint: Use $$ \tan (s+t)=\frac{\sin (s+t)}{\cos (s+t)} $$ and divide the numerator and denominator by \(\cos s \cos t .]\)

Sound Beats When two pure notes that are close in frequency are played together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by $$f_{1}(t)=\cos 11 t \quad \text { and } \quad f_{2}(t)=\cos 13 t$$ the resulting sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph the function \(y=f(t)\) (b) Verify that \(f(t)=2 \cos t \cos 12 t\) (c) Graph \(y=2 \cos t\) and \(y=-2 \cos t,\) together with the graph in part (a), in the same viewing rectangle. How do these graphs describe the variation in the loudness of the sound?

Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by \(f_{1}(t)=C \sin \omega t\) and \(f_{2}(t)=C \sin (\omega t+\alpha) .\) The two sound waves interfere to produce a single sound modeled by the sum of these functions $$ f(t)=C \sin \omega t+C \sin (\omega t+\alpha) $$ (a) Use the Addition Formula for Sine to show that \(f\) can be written in the form \(f(t)=A \sin \omega t+B \cos \omega t,\) where \(A\) and \(B\) are constants that depend on \(\alpha .\) (b) Suppose that \(C=10\) and \(\alpha=\pi / 3 .\) Find constants \(k\) and \(\phi\) so that \(f(t)=k \sin (\omega t+\phi)\)

Verify the identity. $$\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta$$

Prove the identity. $$\tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x}$$
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