/*! 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 7 Find the four smallest positive ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Find the four smallest positive numbers \(\theta\) such that \(\tan \theta=1\).

Short Answer

Expert verified
The four smallest positive angles \(\theta\) with \(\tan \theta = 1\) are: \(\frac{\pi}{4}\), \(\frac{5\pi}{4}\), \(\frac{9\pi}{4}\), and \(\frac{13\pi}{4}\).

Step by step solution

01

Understand the tangent function

The tangent function, \(\tan \theta\), is the ratio of the sine and cosine functions, given by the formula: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). In the unit circle, the tangent function is also the slope of the line connecting the origin and the point on the unit circle associated with the angle. When \(\tan \theta = 1\), this line has a slope of 1, meaning that \(\sin \theta = \cos \theta\).
02

Find the smallest angle with \(\tan \theta = 1\)

To find the smallest angle \(\theta\) for which \(\tan \theta = 1\), we look for where the sine and cosine functions are equal. This occurs in the first quadrant of the unit circle, at the angle \(\theta = \frac{\pi}{4}\). Therefore, the smallest angle with \(\tan \theta = 1\) is \(\theta = \frac{\pi}{4}\).
03

Determine the period of the tangent function

The tangent function is periodic, meaning that after a certain angle, the function will have the same value. The period of the tangent function is the difference in angles where the tangent function repeats itself. Since the tangent function is derived from the sine and cosine functions, we take into account their periods. The sine and cosine functions both have a period of \(2\pi\). However, the tangent function repeats itself every \(\pi\) radians, which can be seen as the graph of the tangent function has a period of \(\pi\). So the period of the tangent function is \(\pi\).
04

Find the next three smallest positive angles with \(\tan \theta = 1\)

Since the tangent function has a period of \(\pi\), we can find the next three smallest positive angles \(\theta\) with \(\tan \theta = 1\) by adding multiples of the period to the smallest angle. The smallest angle we found in step 2 is \(\theta = \frac{\pi}{4}\), so we add the period to get the next three angles: 1. \(\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}\) 2. \(\theta = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4}\) 3. \(\theta = \frac{\pi}{4} + 3\pi = \frac{13\pi}{4}\) These are the next three smallest positive angles with \(\tan \theta = 1\).
05

Write the final solution

The four smallest positive angles with \(\tan \theta = 1\) are as follows: 1. \(\theta = \frac{\pi}{4}\) 2. \(\theta = \frac{5\pi}{4}\) 3. \(\theta = \frac{9\pi}{4}\) 4. \(\theta = \frac{13\pi}{4}\)

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!

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

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\tan (-\gamma)\)

(a) Show that $$x^{3}+x^{2} y+x y^{2}+y^{3}=\left(x^{2}+y^{2}\right)(x+y)$$ for all numbers \(x\) and \(y\). (b) Show that $$ \begin{aligned} \cos ^{3} \theta+\cos ^{2} \theta \sin \theta &+\cos \theta \sin ^{2} \theta+\sin ^{3} \theta \\ &=\cos \theta+\sin \theta \end{aligned} $$

Find the four smallest positive numbers \(\theta\) such that \(\sin \theta=-1\).

Find the smallest positive number \(x\) such that $$ \tan x=3 \tan \left(\frac{\pi}{2}-x\right) $$

Find exact expressions for the indicated quantities. \(\tan \left(\frac{\pi}{2}-v\right)\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.