/*! 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 4 Solve each problem. Find \(\co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 4

Solve each problem. Find \(\cos (\alpha),\) given that \(\sin (\alpha)=0\) and \(\cos (\alpha)<0\).

Short Answer

Expert verified
\cos (180^\backslashcirc) = -1\.

Step by step solution

01

Understanding the Given Information

The problem states that \(\sin (\alpha)=0\), which gives us a specific set of possible angles. It also indicates that \(\cos (\alpha)<0\), narrowing down the possible solutions further.
02

Identify Angles Where \(\sin (\alpha)=0\)

The sine of an angle is 0 at \(\alpha = 0^\backslashcirc\), \(\alpha = 180^\backslashcirc\), \(\text{or } 360^\backslashcirc\). Given the periodicity of trigonometric functions, these angles can be generalized as \(\alpha = k \pi\), where \(k\) is an integer.
03

Determine Where \(\cos (\alpha)

The cosine of an angle is negative in the second and third quadrants. From the above general angles, \(\0^\backslashcirc\) and \(\360^\backslashcirc\) fall on the x-axis where \(\cos (\alpha)>0\), leaving \(\alpha = 180^\backslashcirc\) as the angle where \(\cos (\alpha)<0\).
04

Calculate \(\cos (\alpha)\) for \(\alpha = 180^\backslashcirc\)

\cos (180^\backslashcirc) = -1\. Thus, the value of \(\cos (\alpha)\) is negative and satisfies the given conditions.

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.

Understanding Sin(伪) = 0
In trigonometry, \(\text{sin}(\text{伪}) = 0\) is a significant statement. This tells us that the angle 伪 is at a position where the sine value touches zero.
The sine function reflects the y-coordinate of a point on the unit circle. Hence, when sin(伪) = 0, the unit circle's point lies on the x-axis.
These specific points are at \(\text{伪} = 0^\backslashcirc, 180^\backslashcirc, 360^\backslashcirc\), and their periodic equivalents.
Generally, you can express these angles as \(\text{伪} = k \backslashpi\)(蟺) where k is an integer.
Always remember the periodic nature of trigonometric functions, meaning these values repeat every 360 degrees or 2蟺 radians.
Cos(伪) < 0
Cosine (cos) function represents the x-coordinate of a point on the unit circle.
When \(\text{cos}(\text{伪}) < 0\), the angle 伪 is in a position on the unit circle where the x-coordinate is negative.
Cosine is negative in the second and third quadrants of the unit circle.
So, for 伪 to make \(\text{cos}(\text{伪}) < 0\), it has to lie in these quadrants.
When considering the angles where sin(伪) = 0 as derived earlier, you are left with angles like 180掳 because it falls in the second quadrant.
180 Degrees and Its Implications
In the context of this problem, \(\text{伪} = 180^\backslashcirc\) fits both conditions: \(\text{sin}(\text{伪}) = 0\) and \(\text{cos}(\text{伪}) < 0\).
At 180掳, the sine value is zero because it is on the x-axis.
The cosine value at this point is -1 because the point lies on the negative side of the x-axis.
Therefore, for \(\text{伪} = 180^\backslashcirc\), \(\text{cos}(\text{伪}) = -1\), confirming \(\text{cos}(\text{伪}) < 0\).
This ultimately satisfies the given conditions of both the sine and cosine functions.

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

Use a calculator to evaluate each expression. Round approximate answers to four decimal places. $$ \frac{\sin (\pi / 12)}{\cos (\pi / 12)} $$

Find the radius of the circle in which the given central angle \(\alpha\) intercepts an arc of the given length s. Round to the nearest tenth. $$ \alpha=\pi / 6, s=500 \mathrm{ft} $$

Solve each problem. In each case name the quadrant containing the terminal side of \(\alpha\) a. \(\sin \alpha>0\) and \(\cos \alpha<0\) b. \(\sin \alpha<0\) and \(\cos \alpha>0\) c. \(\tan \alpha>0\) and \(\cos \alpha<0\) d. \(\tan \alpha<0\) and \(\sin \alpha>0\)

Solve each problem. If \(\sin \alpha=0,\) then what is \(\csc \alpha ?\)

True or false? Do not use a calculator. $$ \cos (13 \pi / 12)=-\cos (\pi / 12) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.