Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 13

Simplify \(\sin \theta \cos \theta \tan \theta+\cos ^{2} \theta\)

Short Answer

Expert verified
Answer: The simplified form of the given expression is \(1\).

Step by step solution

01

Express everything in terms of sin and cos

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so we can replace the \(\tan \theta\) in our expression. The expression then becomes: \[ \sin \theta \cos \theta \frac{\sin \theta}{\cos \theta}+\cos ^{2} \theta \]
02

Simplify the expression

Now we can simplify the expression by canceling out terms and rearranging. Indeed: \[ \sin \theta \cos \theta \frac{\sin \theta}{\cos \theta}+\cos ^{2} \theta = \sin \theta \cos \theta \cdot \frac{\sin \theta}{\cancel{\cos \theta}}\cancel{\cos \theta}+\cos ^{2} \theta := \sin^{2} \theta +\cos ^{2} \theta\]
03

Recognize the Pythagorean identity

Recall the Pythagorean identity for sin and cos: \[ \sin^{2} \theta + \cos^{2} \theta = 1\]
04

Final simplification

Since our simplified expression is equal to the Pythagorean identity, the final simplified result is: \[1\]

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Show \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\)

Express \(\frac{1}{2} \cos t+\sin t\) in the form \(A \sin (\omega t-\alpha), \alpha \geq 0\)

An angle \(\beta\) is such that \(\cos \beta>0\) and \(\tan \beta<0\). State the range of possible values of \(\beta\).

Show \(\cos \left(360^{\circ}-\theta\right)=\cos \theta\)

Show \(\tan \left(180^{\circ}-\theta\right)=-\tan \theta\)
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Oscillations

Read Explanation

Scientific Method Physics

Read Explanation

Waves Physics

Read Explanation

Space Physics

Read Explanation

Force

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.

Sign-up for free