/*! 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 2 Be sure that you've familiarized... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. In Exercises \(1-8,\) use the appropriate formula to express each product as a sum or difference. $$\sin 8 x \sin 4 x$$

Short Answer

Expert verified
The expression \(\sin 8x \sin 4x\) as a sum or difference, using the appropriate trigonometric identities, simplifies to \(\frac{1}{2}[\cos(4x) - \cos(12x)]\).

Step by step solution

01

Apply the Product-To-Sum Formulas

Use the formula to transform the expression. In the formula \(\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]\), let \(A = 8x\) and \(B = 4x\).
02

Substitution

Substitute 8x and 4x into the equation: \(\frac{1}{2}[\cos(8x - 4x) - \cos(8x + 4x)]\).
03

Simplify the Equation

The expression will simplify to: \(\frac{1}{2}[\cos(4x) - \cos(12x)]\).

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 fundamental relationships between trigonometric functions. They play a crucial role in simplifying expressions and solving trigonometric equations. In the context of our problem, understanding these identities helps us manipulate trigonometric expressions, especially when converting products of trigonometric functions into sums or differences.

There are several key trigonometric identities you should be familiar with:
  • Pythagorean Identities: For example, \( \sin^2(x) + \cos^2(x) = 1 \).
  • Reciprocal Identities: Such as \( \csc(x) = \frac{1}{\sin(x)} \).
  • Quotient Identities: Like \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
  • Co-Function Identities: For example, \( \sin(90^\circ - x) = \cos(x) \).
Mastering these identities not only helps you understand trigonometric functions better but also lays the groundwork for more complex problems involving transformations such as product-to-sum and sum-to-product formulas.
Sum and Difference Formulas
Sum and difference formulas allow us to find the trigonometric function values of sums or differences of two angles. Understanding these formulas is crucial when dealing with expressions involving multiple angles, just like in the exercise where we applied the product-to-sum formula.

The sum and difference formulas for sine and cosine are as follows:
  • For cosine: \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \).
  • For sine: \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \).

These formulas are invaluable for transforming expressions where angles are added or subtracted. They can help simplify the calculation of trigonometric values for non-standard angles, aiding in both academic and practical applications.
Trigonometric Functions
Trigonometric functions are fundamental in understanding the relationships between the sides and angles of triangles, especially right-angled triangles. They are also essential for modeling periodic phenomena such as sound waves, light waves, and tides. In trigonometry, the primary functions include:
  • \(\sin(x)\): Represents the sine function, defined as the ratio of the opposite side to the hypotenuse.
  • \(\cos(x)\): Represents the cosine function, the ratio of the adjacent side to the hypotenuse.
  • \(\tan(x)\): Represents the tangent function, which is the ratio of the sine function to the cosine function \(\left(\frac{\sin(x)}{\cos(x)}\right)\).
These functions are periodic, with sine and cosine functions having a period of \(2\pi\) and the tangent function a period of \(\pi\). Understanding these basic concepts is crucial for analyzing and interpreting trigonometric expressions in a variety of contexts.

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

Exercises \(166-168\) will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using the cross- products principle to clear fractions from the proportion: $$ \text { If } \frac{a}{b}=\frac{c}{d}, \text { then } a d=b c .(b \neq 0 \text { and } d \neq 0) $$ Round to the nearest tenth. $$\text { Solve for } a: \frac{a}{\sin 46^{\circ}}=\frac{56}{\sin 63^{\circ}}$$

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan x=-5$$

Make Sense? In Exercises \(152-155,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. There are similarities and differences between solving \(4 x+1=3\) and \(4 \sin \theta+1=3:\) In the first equation, \(\mathrm{I}\) need to isolate \(x\) to get the solution. In the trigonometric equation, I need to first isolate \(\sin \theta,\) but then 1 must continue to solve for \(\theta\)

Use this information to solve Exercises \(129-130 .\) Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, y, measured in liters per second, after \(x\) seconds is modeled by $$ y=0.6 \sin \frac{2 \pi}{5} x $$ Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we inhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$7 \sin ^{2} x-1=0$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.