/*! 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 Determine whether the statement ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 77

Determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \(u, \sin 2 u=-2 \sin u \cos u\).

Short Answer

Expert verified
The statement is false. The correct expression according to the identities of trigonometric functions should be \(\sin 2(-u) = -2 \sin u \cos u\). Thus, the sine of double a negative value is equal to the negative of double the product of sine and cosine of the value, not double the positive value.

Step by step solution

01

Understand the provided information

The statement provided states that for a negative number \(u\), \(\sin 2u = -2 \sin u \cos u\) because the sine function is an odd function. The odd function property suggests that for any \(x\), \(\sin(-x) = -\sin(x)\). However, this does not directly apply to the double angle formula.
02

Apply the double angle formula for sine

Apply the known double angle formula for sine, which is \(\sin 2u = 2 \sin u \cos u\). This formula remains valid regardless of the sign of \(u\). For a negative \(u\), this would become \(\sin 2(-u) = 2 \sin(-u) \cos(-u)\). Then, due to the even property of the cosine function, it simplifies to \(\sin 2(-u) = 2 \sin(-u) \cos u\). Finally, applying the sine function's odd property transforms the formula to \(\sin 2(-u) = -2 \sin u \cos u\). However, it is important to note that while the left-hand side changes the argument of \(u\) to \(-u\), the right-hand side does not.
03

Compare with the original statement

The original statement implies that for a negative \(u\), \(\sin 2u = -2 \sin u \cos u\). Comparing this statement with what has been derived, it is clearly incorrect. The correct version should be \(\sin 2(-u) = -2 \sin u \cos u\). So the left-hand side of the statement needs to have the sine function argument as \(-u\) or \(2u\) be negative.

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!

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

The length \(s\) of a shadow cast by a vertical gnomon (a device used to tell time) of height \(h\) when the angle of the sun above the horizon is \(\theta\) can be modeled by the equation \(s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}\) (a) Verify that the expression for \(s\) is equal to \(h \cot \theta\) (b) Use a graphing utility to complete the table. Let \(h=5\) feet. (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is \(90^{\circ} ?\)

Find the area of the triangle having the indicated angle and sides. $$C=120^{\circ}, \quad a=4, \quad b=6$$

In Exercise \(64,\) the Law of Cosines was used to solve a triangle in the two- solution case of SSA. Can the Law of cosines be used to solve the no-solution and single-solution cases of SSA? Explain.

A plane flies 810 miles from Franklin to Centerville with a bearing of \(75^{\circ} .\) Then it flies 648 miles from Centerville to Rosemount with a bearing of \(32^{\circ} .\) Draw a figure that visually represents the situation. Then find the straight-line distance and bearing from Franklin to Rosemount.

The Mach number \(M\) of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle \(\theta\) of the cone by \(\sin (\theta / 2)=1 / M\) (a) Use a half-angle formula to rewrite the equation in terms of \(\cos \theta\). (b) Find the angle \(\theta\) that corresponds to a Mach number of 1. (c) Find the angle \(\theta\) that corresponds to a Mach number of 4.5. (d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and \((\mathrm{c})\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.