/*! 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 24 Simplify the given expressions. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 24

Simplify the given expressions. $$4 \sin \frac{1}{2} x \cos \frac{1}{2} x$$

Short Answer

Expert verified
The expression simplifies to \(2 \sin x\).

Step by step solution

01

Identify the Trigonometric Identity

The expression given, \(4 \sin \frac{1}{2} x \cos \frac{1}{2} x\), can be simplified using a trigonometric identity. The identity \(2 \sin A \cos A = \sin 2A\) allows us to simplify expressions like this. In our problem, \(A = \frac{1}{2}x\).
02

Apply the Identity

Apply the identity to the expression: \(2 \sin \frac{1}{2}x \cos \frac{1}{2}x = \sin\left(x\right)\). Since our original expression has a factor of 4, express the original equation as \(2 \times 2 \sin \frac{1}{2}x \cos \frac{1}{2}x = 2 \sin x\).
03

Simplify the Expression

Thus, by applying the identity, the expression simplifies to \(2 \sin x\).

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.

Simplification
Simplifying trigonometric expressions can often make them easier to understand or work with in larger problems. This process involves using known mathematical rules or identities to rewrite an expression in a simpler or more easily used form. In our example, we started with the expression \(4 \sin \frac{1}{2} x \cos \frac{1}{2} x\). By recognizing a applicable trigonometric identity, we were able to simplify it efficiently.

This simplification helps us see the problem from a different angle, often leading to recognizing patterns or solutions more swiftly. In practice, it allows us to focus less on complex calculations and more on solving the main problem.
Sin Double Angle Formula
The formula \(2 \sin A \cos A = \sin(2A)\) is a key trigonometric identity known as the sine double angle formula. This useful tool helps us transform products of sine and cosine into a single sine term. Recognizing when to apply this formula can significantly simplify computations in trigonometry.

In the given exercise, A was set to \(\frac{1}{2}x\), allowing the expression \(2 \sin \frac{1}{2}x \cos \frac{1}{2}x\) to simplify to \(\sin(x)\). This identity is very versatile, making it valuable in both theoretical work and practical calculations.
  • Key Feature: Converts product to a single trigonometric function.
  • Usefulness: Helps in solving trigonometric equations more easily.
Basic Trigonometry
Basic trigonometry involves the study of the relationships between angles and sides of triangles. Fundamental concepts like sine, cosine, and tangent are essential and have numerous applications in fields like physics, engineering, and more. In this exercise, we deal with the sine and cosine functions, which are crucial in understanding the rotational and oscillatory motion.

The sine function, denoted \(\sin\), relates the opposite side of a right triangle to its hypotenuse. Cosine, \(\cos\), does the same for the adjacent side. Mastery of these basic functions allows you to interpret more complex identities and formulas, forming a strong foundation in trigonometry.
  • Importance: Forms the basis of all trigonometric equations and identities.
  • Application: Used extensively in science to solve real-world problems involving triangles.

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

Evaluate the given expressions. In Exercises 61 and \(62,\) find an equivalent algebraic expression. $$\cos \left(\sin ^{-1} \frac{\sqrt{2}}{2}+\cos ^{-1} \frac{3}{5}\right)$$

Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. $$\tan \left(\frac{3 \pi}{4}+x\right)=\frac{\tan x-1}{\tan x+1}$$

Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. $$2 \csc 2 x \tan x=\sec ^{2} x$$

Simplify the given expressions. $$\cos (x+\pi) \cos (x-\pi)+\sin (x+\pi) \sin (x-\pi)$$

Solve the given problems. The design of a certain three-phase alternating-current generator uses the fact that the sum of the currents \(I \cos \left(\theta+30^{\circ}\right), I \cos \left(\theta+150^{\circ}\right),\) and \(I \cos \left(\theta+270^{\circ}\right)\) is zero. Verify this.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.