/*! 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 31 Find the values of the six trigo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 31

Find the values of the six trigonometric functions of \(\theta\) with the given constraint. Function Value \(\qquad\) Constraint \(\cot \theta\) is undefined. \(\quad \pi / 2 \leq \theta \leq 3 \pi / 2\)

Short Answer

Expert verified
The six trigonometric functions values for the given constraint are \(\sin \theta = 0\), \(\cos \theta = -1\), \(\tan \theta = 0\), \(\csc \theta\) is undefined, \(\sec \theta = -1\), and \(\cot \theta\) is undefined.

Step by step solution

01

Identify the Accurate Angle

Since \(\cot \theta\) is undefined whenever \(\sin \theta = 0\) and the denominator \(\sin \theta\) does not equal zero. Query on what angle between \(\pi / 2\) and \(3 \pi / 2\) has its sine equals to zero. From basic knowledge of trigonometric functions, \(\sin \theta = 0\) at \(\theta = \pi\).
02

Find the Values of Trigonometric Functions

Now as \(\theta = \pi\) is identified, just substitute \(\theta\) with \(\pi\) in the definitions of the six trigonometric functions to get their values in this particular case. For \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\), their values are \(0\), \(-1\), and \(0\) respectively. For \(\csc \theta\), \(\sec \theta\), and \(\cot \theta\), since they are the reciprocal of the first three, their values are 'undefined', \(-1\), and 'undefined' respectively since we can't divide by zero.
03

Formulate Final Results

The results of the six trigonometric functions of \(\theta\) in our case are \(\sin \theta = 0\), \(\cos \theta = -1\), \(\tan \theta = 0\), \(\csc \theta\) is undefined, \(\sec \theta = -1\), and \(\cot \theta\) is undefined.

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.

Cotangent
Cotangent, written as \(\cot \theta\), is one of the six main trigonometric functions. It is known as the reciprocal of the tangent function. That means it is defined as:
  • \(\cot \theta = \frac{1}{\tan \theta}\)
  • Which is also \(\frac{\cos \theta}{\sin \theta}\)
The cotangent function is particularly interesting because it is undefined at angles where the sine of \(\theta\) is zero. This is because you cannot divide by zero, and it leads to an undefined expression. For example, as mentioned in the exercise, \(\cot \theta\) becomes undefined at \(\theta = \pi\) due to \(\sin \theta = 0\). This is a critical point when analyzing the behavior of trigonometric functions.
Sine and Cosine
The sine and cosine functions, \(\sin \theta\) and \(\cos \theta\), are fundamental in trigonometry. They help describe the relationship between the angle \(\theta\) and the ratios of sides in right-angled triangles. Typically, sine represents the ratio of the opposite side to the hypotenuse, while cosine corresponds to the adjacent side to the hypotenuse in a right triangle.
  • For \(\theta = \pi\), we find \(\sin \theta = 0\)
  • \(\cos \theta = -1\)
The angle \(\theta = \pi\) lies on the negative x-axis of the unit circle. This placement of the angle results in specific values for sine and cosine, and both these values are critical in evaluating the values of the other trigonometric functions.
Reciprocal Identities
Reciprocal identities connect a trigonometric function to its reciprocal counterpart. They are a vital tool for simplifying and solving trigonometric equations:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
From this, if \(\sin \theta = 0\), \(\csc \theta\) becomes undefined, since you cannot divide by zero. Similarly, when \(\cos \theta = 0\), \(\sec \theta\) would be undefined. Therefore, understanding these reciprocal relationships helps one predict when particular functions become undefined, which is very valuable for students analyzing trigonometric expressions.
Unit Circle
The unit circle is a crucial concept in trigonometry. It is a circle with a radius of one centered at the origin of a coordinate plane. This circle provides a geometric way of understanding angles and their corresponding sine and cosine values.
  • Every angle \(\theta\) on the unit circle can be represented as a point \((\cos \theta, \sin \theta)\)
  • At \(\theta = \pi\), this point is \((-1, 0)\)
This correspondence helps in visualizing the periodic nature and key characteristics of trigonometric functions. It also aids in understanding why certain functions such as \(\cot \theta\) or \(\csc \theta\) become undefined at particular angles, such as when \(\theta\) equals multiples of \(\pi\). Knowing the unit circle and its components enables a deeper comprehension of trigonometric identities and their applications.

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

Find two solutions of each equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi) .\) Do not use a calculator. (a) \(\tan \theta=1\) (b) \(\cot \theta=-\sqrt{3}\)

Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by \(y(t)=2 e^{-t} \cos 6 t,\) where \(y\) is the displacement (in centimeters) and \(t\) is the time (in seconds). Find the displacement when (a) \(t=0,\) (b) \(t=\frac{1}{4},\) and (c) \(t=\frac{1}{2}\)

A ship leaves port at noon and has a bearing of \(\mathrm{S} 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6: 00 P.M.? (b) At 6: 00 P.M., the ship changes course to due west. Find the ship's bearing and distance from the port of departure at 7: 00 P.M.

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{2} \cos 20 \pi t$$

Write the function in terms of the sine function by using the identity. $$A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right)$$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$f(t)=3 \cos 2 t+3 \sin 2 t$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.