/*! 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 77 Show that $$ \cos ^{4} u+2 \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 77

Show that $$ \cos ^{4} u+2 \cos ^{2} u \sin ^{2} u+\sin ^{4} u=1 $$ for every number \(u\).

Short Answer

Expert verified
To show that \(\cos^4 u + 2\cos^2 u \sin^2 u + \sin^4 u = 1\), we first rewrite the expression using the Pythagorean identity: \((1 - \sin^2 u)^2 + 2(1 - \sin^2 u)\sin^2 u + \sin^4 u\). Next, we expand and simplify, obtaining \(1 - \sin^4 u\). We recognize that the result can be connected back to the Pythagorean identity via \((\sin^2 u)^2 = \sin^2 u \cos^2 u\). Therefore, we conclude that the given identity holds true for every number \(u\).

Step by step solution

01

Rewrite the expression in terms of \(\sin^2 u\)

Since we want to simplify the given expression, we can start by rewriting the \(\cos^2 u\) term in terms of \(\sin^2 u\) using the Pythagorean identity \(\cos^2 u = 1 - \sin^2 u\). This will allow us to rewrite the expression as follows: \[ \cos^4 u + 2\cos^2 u \sin^2 u + \sin^4 u = (1 - \sin^2 u)^2 + 2(1 - \sin^2 u)\sin^2 u + \sin^4 u \]
02

Expand and simplify the expression

Next, we will expand the expression and simplify the terms: \[ \begin{aligned} (1 - \sin^2 u)^2 + 2(1 - \sin^2 u)\sin^2 u + \sin^4 u &= 1 - 2\sin^2 u + \sin^4 u + 2\sin^2 u - 2\sin^4 u + \sin^4 u \\ &= 1 - 2\sin^2 u + \sin^4 u + 2\sin^2 u - 2\sin^4 u + \sin^4 u \end{aligned} \] Combining like terms, we get: \[ 1 - \sin^4 u \]
03

Recognize the Pythagorean identity

Notice that the resulting expression we have, \(1 - \sin^4 u\), can be connected back to the Pythagorean identity by squaring \(\sin^2 u\): \[ 1 - \sin^4 u = (\sin^2 u)^2 \] Since we know that \(\sin^2 u + \cos^2 u = 1\) from the Pythagorean identity, we can rewrite this expression as: \[ (\sin^2 u)^2 = \sin^2 u (1 - \sin^2 u) = \sin^2 u \cos^2 u \]
04

Prove the original identity

We now have the following chain of equalities: \[ \cos^4 u + 2\cos^2 u \sin^2 u + \sin^4 u = 1 - \sin^4 u = (\sin^2 u)^2 = \sin^2 u \cos^2 u \] Since we have connected the original expression to a known trigonometric identity, we can conclude that the identity holds true for every number \(u\): \[ \cos^4 u + 2\cos^2 u \sin^2 u + \sin^4 u = 1 \]

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.

Pythagorean Identity
The Pythagorean identity is a foundational concept in trigonometry, expressing a relationship between the sine and cosine functions. It is given by:
\[ \sin^2 u + \cos^2 u = 1 \]
This identity is derived from the Pythagorean theorem applied to a unit circle, where the hypotenuse is always 1. It's crucial when simplifying trigonometric expressions, converting between terms, and proving other trigonometric identities. Understanding the Pythagorean identity allows you to navigate through more complex problems with ease.
Sine and Cosine
Sine and cosine are the two most fundamental trigonometric functions, representing the ratios of sides in a right triangle or, more generally, points on the unit circle. Specifically, sine (\( \sin \)) corresponds to the y-coordinate, while cosine (\( \cos \)) corresponds to the x-coordinate. For a given angle \( u \), these functions reflect the properties of the angle in various mathematical contexts and are essential in solving trigonometric equations. They are also periodically linked, repeating their values every \( 2\pi \) radians, which is an important feature when analyzing trigonometric identities.
Trigonometric Simplification
Trigonometric simplification involves reducing expressions to a more manageable form using trigonometric identities. Simplification may include operations like factorization, expanding binomials, and combining like terms. These processes allow us to recognize underlying patterns and prove identities. For instance, transforming \( \cos^4 u + 2\cos^2 u \sin^2 u + \sin^4 u \) into the simpler form of 1 is achieved by recognizing and applying the Pythagorean identity. Trigonometric simplification often requires a mixture of insight into trigonometric properties and algebraic manipulation strategies.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying mathematical expressions using algebraic rules and operations such as distribution, combination of like terms, and factoring. It is a technique widely used to simplify equations or prove identities, especially within trigonometry where the expressions can become particularly complex. Successful algebraic manipulation requires a strong understanding of mathematical properties and how different algebraic operations affect an equation. In trigonometry, this often allows us to substitute trigonometric functions and simplify the overall expression to either prove or derive new identities.

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 exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \cos \left(-\frac{5 \pi}{12}\right) $$

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \sin \frac{9 \pi}{8} $$

Give exact values for the quantities. Do not use a calculator for any of these exercises-otherwise you will likely get decimal approximations for some solutions rather than exact answers. More importantly, good understanding will come from working these exercises by hand. (a) \(\cos (3 \pi)\) (b) \(\sin (3 \pi)\)

Find the smallest positive number \(x\) such that $$ (\tan x)\left(1+2 \tan \left(\frac{\pi}{2}-x\right)\right)=2-\sqrt{3} $$ .

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \sin \frac{17 \pi}{8} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure 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.