/*! 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 864 Change \(\tan \theta(\sin \theta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 27: Problem 864

Change \(\tan \theta(\sin \theta+\cot \theta \cos \theta)\) to \(\sec \theta\)

Short Answer

Expert verified
We can change the expression \(\tan \theta(\sin \theta+\cot \theta \cos \theta)\) to \(\sec \theta\) by using trigonometric identities and relationships. First, rewrite \(\cot \theta\) and \(\tan \theta\) in terms of sine and cosine as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute these back into the original expression and simplify to obtain \(\frac{\sin^2\theta}{\cos\theta}+1\). Use the Pythagorean identity to rewrite this as \(\frac{1-\cos^2\theta}{\cos\theta}+1\), which, after combining the fractions, simplifies to \(\frac{1}{\cos\theta}\), or \(\sec\theta\).

Step by step solution

01

Rewrite cotangent in terms of sine and cosine

Rewrite the cotangent term in the expression as the ratio of cosine to sine: \[\cot \theta = \frac{\cos \theta}{\sin \theta}\]
02

Substitute cotangent in the given expression

Now substitute the expression for \(\cot \theta\) that we found in step 1 back into the original expression: \[\tan \theta(\sin \theta+\frac{\cos^2 \theta}{\sin \theta})\]
03

Rewrite tangent in terms of sine and cosine

Rewrite the tangent term as the ratio of sine to cosine: \[\tan \theta = \frac{\sin \theta}{\cos \theta}\]
04

Substitute tangent in the expression

Now substitute the expression for \(\tan \theta\) that we found in step 3 back into the expression from step 2: \[\frac{\sin \theta}{\cos \theta}(\sin \theta+\frac{\cos^2 \theta}{\sin \theta})\]
05

Simplify the expression

Multiply the terms inside the parenthesis by the sine over cosine term: \[\frac{\sin^2\theta}{\cos \theta}+\frac{\cos^2\theta \sin\theta}{\sin\theta \cos\theta}\]
06

Simplify the expression further

Now, simplify the second term by cancelling out the sine term in the numerator and denominator: \[\frac{\sin^2 \theta}{\cos \theta} + \frac{\cos \theta}{\cos \theta}\]
07

Add the terms

Now, observe that the second term is equal to \(1\). With that in mind, add the terms together: \[\frac{\sin^2\theta}{\cos\theta}+1\]
08

Rewrite using Pythagorean identity

Use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) to rewrite the expression as: \[\frac{1-\cos^2\theta}{\cos\theta}+1\]
09

Combine the fractions

Combine the fractions by finding a common denominator, which in this case is \(\cos\theta\): \[\frac{1-\cos^2\theta+\cos^2\theta}{\cos\theta}\]
10

Simplify to obtain secant

Observe that the numerator simplifies to \(1\). The expression becomes: \[\frac{1}{\cos\theta}\] Now we can see that the resulting expression is the reciprocal of the cosine function, which is the secant function: \[\sec\theta\] Thus, we have successfully changed the given expression \(\tan \theta(\sin \theta+\cot \theta \cos \theta)\) to \(\sec \theta\).

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 triangle \(A B C\), given \(a=30, b=50, \angle C=25^{\circ}\)

If \(\tan \theta=3.8436\), find \(\theta\).

What primary angle is conterminal with the angle of \(-743^{\circ}\) ?

Find \(\cos (1 / 12) \pi\)

Find \(\sin 202(1 / 2)^{\circ}\) and \(\tan 202(1 / 2)^{\circ}\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.