/*! 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 247 For the following exercises, fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 247

For the following exercises, find the exact value. $$ \cos ^{-1}(\sin (\pi)) $$

Short Answer

Expert verified
The exact value is \(\pi/2\).

Step by step solution

01

Evaluate the inner function

First, evaluate the expression inside the parentheses, \( \sin(\pi) \). From trigonometric identities, we know that \( \sin(\pi) = 0 \). This simplifies the expression to \( \cos^{-1}(0) \).
02

Determine the angle for \( \cos^{-1}(0) \)

Next, find the angle whose cosine value is 0. The inverse cosine function \( \cos^{-1}(x) \) gives the angle in the principal range \([0, \pi]\). In this range, \( \cos(\pi/2) = 0 \). Therefore, \( \cos^{-1}(0) = \pi/2 \).

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.

Trigonometric Identities
Trigonometric identities are equations that are true for all values of the involved variables, wherever the expressions are defined. They help simplify complex trigonometric expressions. One of the key identities to remember is the
  • Sine of 0 degrees and 180 degrees, which are 0.
  • Cosine of 90 degrees, which is also 0.
These identities are not only helpful in simplifying expressions but are crucial in solving trigonometric equations too. In our original exercise, we used the identity \(\sin(\pi) = 0\) to simplify the problem. Recognizing these trigonometric identities allows us to evaluate functions accurately and efficiently.
Inverse Cosine Function
The inverse cosine function, denoted as \(\cos^{-1}(x)\), is a function used to find angles when the cosine value is known. Inverse trigonometric functions can reverse the action of trigonometric functions, with specific ranges. For \(\cos^{-1}(x)\), this function provides an angle whose cosine is \(x\). The range of this function is particularly important because:
  • It only returns angles within \([0, \pi]\).
  • This range is part of ensuring that each cosine input maps to a unique angle output.
In problems like our exercise, where we start with a cosine value (like 0), \(\cos^{-1}(x)\) can directly give us the corresponding angle in this range, like \(\frac{\pi}{2}\) for \(\cos^{-1}(0)\). Understanding this function is key to solving inverse trigonometry problems effectively.
Principal Range of Cosine
The principal range of cosine is a concept that defines the range of angles that the inverse cosine function will consider. This ensures the function only provides valid outputs. For cosine, this principal range is
  • \([0, \pi]\), which contains all possible angles where cosine values transition from 1 to -1.
  • This makes it unique in trigonometric functions since it only includes angles from the first and second quadrants.
Knowing about the principal range is crucial because it allows us to specify the correct angle when calculating using the inverse cosine function. For instance, if \(\cos^{-1}(0)\) is calculated, this principal range tells us the answer has to be within \([0, \pi]\), leading us to \(\frac{\pi}{2}\). The principal range is part of what ensures consistent and accurate problem solving in trigonometry.

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

For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes. $$ f(x)=4 \csc (5 x) $$

For the following exercises, find the exact value without the aid of a calculator. Graph \(f(x)=\cos x\) and \(f(x)=\sec x\) on the interval \([0,2 \pi)\) and explain any observations.

For the following exercises, find the exact value without the aid of a calculator. $$ \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

For the following exercises, graph the functions on the specified window and answer the questions. Graph \(m(x)=\sin (2 x)+\cos (3 x)\) on the viewing window \([-10,10]\) by \([-3,3] .\) Approximate the graph's period.

What perentage grade should a road have if elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100 -foot horizontal distance. For example a 5\(\%\) grade means that the road rises 5 feet for every 10 feet of horizontal distance.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.