/*! 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 6 Use a process similar to the one... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Use a process similar to the one in exercise (5) to complete each of the following: (a) Suppose it is known that \(-\frac{1}{4}<\sin (t)<0\) and that \(\cos (t)>0 .\) What can be concluded about \(\cos (t) ?\) (b) Suppose it is known that \(0 \leq \sin (t) \leq \frac{3}{7}\) and that \(\cos (t)<0 .\) What can be concluded about \(\cos (t) ?\)

Short Answer

Expert verified
Given the information about sin(t), (a) we can conclude that \(0 < \cos(t) < \sqrt{\frac{5}{4}}\) when \(-\frac{1}{4}<\sin (t)<0\) and \(\cos (t)>0\). (b) we can conclude that \(-\sqrt{\frac{40}{49}} \leq \cos(t) < 0\) when \(0 \leq \sin (t) \leq \frac{3}{7}\) and \(\cos(t)<0\).

Step by step solution

01

(a) Finding the range of cos(t) given sin(t) information

First, we have the information that \(-\frac{1}{4}<\sin (t)<0\) and \(\cos (t)>0\). We know that in the unit circle, if sin(t) is negative, then the angle t lies in the third or fourth quadrant. Since \(\cos (t)>0\), we can determine that the angle t lies in the fourth quadrant, as cosine is positive in that quadrant. Now we need to find the range of cos(t). We know that \(-\frac{1}{4}<\sin (t)<0\). Using the Pythagorean identity \(\sin^2(t)+\cos^2(t)=1\), we get: \(1-\cos^2(t) > -\frac{1}{4}\) \(\cos^2(t) < 1 + \frac{1}{4}\) \(\cos^2(t) < \frac{5}{4}\) Since the angle t is in the fourth quadrant and cosine is positive in that quadrant, we get: \(0 < \cos(t) < \sqrt{\frac{5}{4}}\) So, given the information about sin(t), we can conclude that \(0 < \cos(t) < \sqrt{\frac{5}{4}}\).
02

(b) Finding the range of cos(t) given sin(t) information

For this case, we have the information that \(0 \leq \sin (t) \leq \frac{3}{7}\) and \(\cos (t)<0\). Knowing sin(t) is non-negative, we can determine that the angle t lies in the first or second quadrant. Since \(\cos (t)<0\), we can determine that the angle t lies in the second quadrant, as cosine is negative in that quadrant. Now we need to find the range of cos(t). We know that \(0 \leq \sin (t) \leq \frac{3}{7}\). Using the Pythagorean identity \(\sin^2(t)+\cos^2(t)=1\), we get: \(1-\cos^2(t) \leq \frac{9}{49}\) \(\cos^2(t) \geq 1 - \frac{9}{49}\) \(\cos^2(t) \geq \frac{40}{49}\) Since the angle t is in the second quadrant and cosine is negative in that quadrant, we get: \(-\sqrt{\frac{40}{49}} \leq \cos(t) < 0\) So, given the information about sin(t), we can conclude that \(-\sqrt{\frac{40}{49}} \leq \cos(t) < 0\).

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.

Quadrants
In trigonometry, understanding the concept of quadrants is vital because it affects the sign of trigonometric functions like sine and cosine. The unit circle, divided into four quadrants, provides a geometric representation that helps us comprehend these relationships.
The quadrants start from the positive x-axis and go counter-clockwise:
  • First Quadrant (0 to 90 degrees): Both sine and cosine are positive.
  • Second Quadrant (90 to 180 degrees): Sine is positive, but cosine is negative.
  • Third Quadrant (180 to 270 degrees): Both sine and cosine are negative.
  • Fourth Quadrant (270 to 360 degrees): Sine is negative, but cosine is positive.
Recognizing which quadrant an angle lies in allows us to determine the signs of trigonometric functions. For example, in the exercise, knowing that \(-\frac{1}{4}<\sin (t)<0\) tells us that angle t must be in the third or fourth quadrant. The additional information that \(\cos (t)>0\) helps us narrow it down to the fourth quadrant.
Pythagorean Identity
The Pythagorean Identity is a fundamental relationship in trigonometry that connects sine and cosine. This identity asserts \(\sin^2(t) + \cos^2(t) = 1\) for any angle \(t\). This equation stems from the Pythagorean Theorem applied to the unit circle.
The identity is incredibly useful in solving trigonometric equations and finding unknown values. For instance, in the exercise, we use this identity to calculate the bounds of \(\cos(t)\). Given that \(-\frac{1}{4} < \sin (t) < 0\), substituting \(\sin(t)\) into the identity gives:
  • \(1 - \cos^2(t) > -\frac{1}{4}\)
  • \(\cos^2(t) < \frac{5}{4}\)
Because \(\cos(t)\) is positive in the fourth quadrant, we determine \(0 < \cos(t) < \sqrt{\frac{5}{4}}\). This identity helps us bridge the gap between known and unknown trigonometric values.
Trigonometric Inequalities
Trigonometric inequalities play a crucial role when determining the possible values of trigonometric functions given certain conditions. These inequalities often leverage the quadrant and identity rules we discussed earlier to narrow down the range of possible function values.
Using the exercise as an example, the inequality \(-\frac{1}{4} < \sin(t) < 0\) implies that sine is a small negative number, hinting at angles near the x-axis in either the third or fourth quadrant. Simultaneously, the condition \(\cos(t) > 0\) confirms that the angle lies in the fourth quadrant.
  • In the first part of the exercise, these inequalities helped us determine bounded values of \(\cos(t)\).
  • In the second part, inequalities \(0 \leq \sin(t) \leq \frac{3}{7}\) and \(\cos(t) < 0\) indicated that the angle is in the second quadrant, leading us to find \(\cos(t)\) in the range \(-\sqrt{\frac{40}{49}} \leq \cos(t) < 0\).
This process shows how mathematical inequalities can accurately restrict or verify the possible scenarios, helping solve trigonometric problems efficiently.

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

Determine the arc length (to the nearest hundredth of a unit when necessary) for each of the following. (a) An arc on a circle of radius 6 feet that is intercepted by a central angle of \(\frac{2 \pi}{3}\) radians. Compare this to one-third of the circumference of the circle. (b) An arc on a circle of radius 100 miles that is intercepted by a central angle of 2 radians. (c) An arc on a circle of radius 20 meters that is intercepted by a central angle of \(\frac{13 \pi}{10}\) radians. (d) An arc on a circle of radius 10 feet that is intercepted by a central angle of 152 degrees.

Suppose it is known that \(0<\cos (t)<\frac{1}{3}\) (a) By squaring the expressions in the given inequalities, what conclusions can be made about \(\cos ^{2}(t) ?\) (b) Use part (a) to write inequalities involving \(-\cos ^{2}(t)\) and then inequalities involving \(1-\cos ^{2}(t)\) (c) Using the Pythagorean identity, we see that \(\sin ^{2}(t)=1-\cos ^{2}(t)\). Write the last inequality in part (b) in terms of \(\sin ^{2}(t)\). (d) If we also know that \(\sin (t)>0,\) what can we now conclude about the value of \(\sin (t) ?\)

In each of the following, when it is possible, determine the exact measure of the central angle in radians. Otherwise, round to the nearest hundredth of a radian. (a) The central angle that intercepts an arc of length \(3 \pi\) feet on a circle of radius 5 feet. (b) The central angle that intercepts an arc of length 18 feet on a circle of radius 5 feet. (c) The central angle that intercepts an arc of length 20 meters on a circle of radius 12 meters.

Determine the quadrant in which the terminal point of each arc lies based on the given information. (a) \(\cos (x)>0\) and \(\tan (x)<0\) (b) \(\tan (x)>0\) and \(\csc (x)<0\) (c) \(\cot (x)>0\) and \(\sec (x)>0\) (d) \(\sin (x)<0\) and \(\sec (x)>0\) (e) \(\sec (x)<0\) and \(\csc (x)>0\) (f) \(\sin (x)<0\) and \(\cot (x)>0\)

Draw the following arcs on the unit circle. (a) The arc that is determined by the interval \(\left[0, \frac{\pi}{6}\right]\) on the number line. (b) The arc that is determined by the interval \(\left[0, \frac{7 \pi}{6}\right]\) on the number line. (c) The arc that is determined by the interval \(\left[0,-\frac{\pi}{3}\right]\) on the number line. (d) The arc that is determined by the interval \(\left[0,-\frac{4 \pi}{5}\right]\) on the number line.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.