/*! 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 42 Use the fundamental identities t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 42

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\frac{\cos ^{2} y}{1-\sin y}$$

Short Answer

Expert verified
The simplified form of the expression \(\frac{\cos ^{2} y}{1-\sin y}\) is \(1 + \sin y\)

Step by step solution

01

Recognize the Pythagorean identity

Start by recognizing that \(\cos^{2}(y)\) is part of the Pythagorean identity \(\sin^{2}(y) + \cos^{2}(y) = 1\). Therefore, It is possible to rewrite \(\cos^{2}(y)\) in terms of \(\sin(y)\), i.e, \(\cos^{2}(y) = 1 - \sin^{2}(y)\).
02

Use the Pythagorean identity

Substitute the expression of \(\cos^{2}(y)\) into our original expression. So it becomes \(\frac{1-\sin^{2} y}{1-\sin y}\).
03

Simplify the expression

Here is where we recognize that the denominator could be factored out of the numerator. (1-\(\sin^{2} y\)) is equivalent to (1-\(\sin y\)) \((1+\sin y)\). So the expression becomes: \(\frac{(1-\sin y)(1+\sin y)}{1-\sin y}\).
04

Simplify further the fraction

In the expression from step 3, we see that (1-\(\sin y\)) appears in the numerator and the denominator. Therefore, they cancel each other. Thus the expression simplifies to (1+\(\sin y\))
05

Final verification

Always check if the result could be simplified. The expression (1+\(\sin y\)) is the simplest form. Therefore, we have completed.

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

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\). $$\sin \left(x+\frac{\pi}{2}\right)+\cos ^{2} x=0$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\tan \frac{x}{2}-\sin x=0$$

Write the expression as the sine, cosine, or tangent of an angle. $$\cos 130^{\circ} \cos 40^{\circ}-\sin 130^{\circ} \sin 40^{\circ}$$

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing window. Use the graphs to determine whether \(y_{1}=y_{2}\) Explain your reasoning. $$y_{1}=\sin (x+4), \quad y_{2}=\sin x+\sin 4$$

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

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.