/*! 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 3 Complete each statement, or answ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

Complete each statement, or answer the question. \(y=\cos ^{-1} x\) means that \(x=\) _____, for \(0 \leq y \leq \pi\).

Short Answer

Expert verified
x = \cos(y)

Step by step solution

01

Identify the given equation

The given equation is in the form of an inverse trigonometric function: \[ y = \cos^{-1} x \].
02

Understand the meaning of the inverse cosine function

The equation \( y = \cos^{-1} x \) implies that \( y \) is the angle whose cosine is \( x \).
03

Express the relationship

The next step is to express the relationship in terms of \( x \). From the definition, if \( y = \cos^{-1} x \), then it means \( \cos(y) = x \).
04

Specify the domain and range

It is given that the range of \( y \) is \( 0 \leq y \leq \pi \), which are the typical bounds for the inverse cosine function to produce real values.
05

Combine to form final equation

Putting all the information together, if \( y = \cos^{-1} x \), then \( x = \cos(y) \) for the range \( 0 \leq y \leq \pi \).

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.

Inverse Cosine
The inverse cosine function, denoted as \(\text{cos}^{-1}(x)\), is used to find the angle whose cosine value is \(x\). This is essential in trigonometry when you need to work backwards from a cosine value to find the angle.

The notation \(y = \text{cos}^{-1}(x)\) means that \(\text{cos}(y) = x\). In simple terms, if you know \(\text{cos}(30^\text{掳}) = 0.866\), then \( \text{cos}^{-1}(0.866) = 30^\text{掳}\).

Remember, inverse trigonometric functions are designed to work within specific ranges for their outputs.
Domain and Range
Understanding the domain and range of trigonometric functions is crucial. For the inverse cosine function, \(y = \text{cos}^{-1}(x)\),

  • Domain: The set of all possible input values (x), which is \([-1, 1]\). This is because the cosine function can only produce values within this interval.
  • Range: The set of all possible output values (y), which is \([0, \pi]\) (in radians). This range ensures the function produces unique (one-to-one) and real results.

For instance, \( \text{cos}^{-1}(1) = 0\) and \( \text{cos}^{-1}(-1) = \pi\). Understanding these constraints is vital when solving problems involving inverse cosine and ensures meaningful solutions.
Trigonometric Relationships
It's important to understand how the inverse cosine function fits into broader trigonometric relationships. These relationships help solve complex problems involving angles and their trigonometric ratios.

  • The fundamental relationship here is \(\text{cos}(y) = x\) when \(y = \text{cos}^{-1}(x)\).
  • This means, to find an angle \(y\) knowing \(x\), we use \( \text{cos}^{-1}(x)\).

For example, if \( \text{cos}(y) = 0.5\), then \( y = \text{cos}^{-1}(0.5) \Rightarrow y = \pi/3 \text{ or } 60^\text{掳} \).
These relationships make it easier to translate between different forms (angles to ratios and vice versa) and solve a variety of 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

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \frac{\theta}{2}=1$$

Amperage, Wattage, and Voltage Amperage is a measure of the amount of electricity that is moving through a circuit, whereas voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of the amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.)(a) A household circuit has voltage $$ V=163 \sin 120 \pi t $$ when an incandescent light bulb is turned on with amperage $$ I=1.23 \sin 120 \pi t $$ Graph the wattage \(W=V I\) consumed by the light bulb in the window \([0,0.05]\) by $$ [-50,300] $$ (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to determine values for \(a, c,\) and \(\omega\) so that \(W=a \cos (\omega t)+c\) (d) Check your answer by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph to estimate the average wattage used by the light. For how many watts (to the nearest integer) do you think this incandescent light bulb is rated?

Verify that each trigonometric equation is an identity. $$\sec x-\cos x+\csc x-\sin x-\sin x \tan x=\cos x \cot x$$

Use a calculator to find each value. Give answers as real numbers. $$\cot (\arccos 0.58236841)$$

The following function approximates the average monthly temperature \(y\) (in \(^{\circ} \mathrm{F}\) ) in Phoenix, Arizona. Here \(x\) represents the month, where \(x=1\) corresponds to January, \(x=2\) corresponds to February, and so on. $$ f(x)=19.5 \cos \left[\frac{\pi}{6}(x-7)\right]+70.5 $$ When is the average monthly temperature (a) \(70.5^{\circ} \mathrm{F}\) (b) \(55^{\circ} \mathrm{F} ?\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

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.