/*! 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 55 Use the fundamental identities t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 55

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\sin \beta \tan \beta+\cos \beta$$

Short Answer

Expert verified
The simplified form of the expression \(\sin \beta \tan \beta + \cos \beta\) could be \(\sin^2 \beta + \cos \beta\) or \(1 - \cos^2 \beta + \cos \beta\).

Step by step solution

01

Recognize the Fundamental Identities

The fundamental identities include: 1) Pythagorean Identity: \(\sin^2 \beta + \cos^2 \beta = 1\) 2) Quotient Identities: \(\tan \beta = \sin \beta / \cos \beta\), and \(\cot \beta = \cos \beta / \sin \beta\)
02

Substitute the identities

Next, we substitute the \(\tan \beta\) in the expression with its equivalent identity. So our original expression \(\sin \beta \tan \beta+\cos \beta\) becomes \(\sin \beta * (\sin \beta / \cos \beta) + \cos \beta\)
03

Simplify the expression

Simplify the expression by performing the indicated operation. We end up with \(\sin^2 \beta + \cos \beta\)
04

Further Simplify using Pythagorean Identity

The expression \(\sin^2 \beta + \cos \beta\) could further be simplified if we recognize that \(\sin^2 \beta = 1 - \cos^2 \beta\) (from Pythagorean Identity). Substituting into the expression gives us \(1 - \cos^2 \beta + \cos \beta\)

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

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 140^{\circ}-\tan 60^{\circ}}{1+\tan 140^{\circ} \tan 60^{\circ}}$$

Find the exact value of the expression. $$\frac{\tan (5 \pi / 6)-\tan (\pi / 6)}{1+\tan (5 \pi / 6) \tan (\pi / 6)}$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$\frac{\cos x \cot x}{1-\sin x}=3$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.