/*! 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 7 Write each expression in terms o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 7

Write each expression in terms of sines and/or cosines, and then simplify. \(\tan x \cos x\)

Short Answer

Expert verified
sin x

Step by step solution

01

Use the definition of tangent

Recall that \(\tan x = \frac{\text{sin}x}{\text{cos}x}\). Substitute this definition into the given expression: \(\tan x \text{cos} x = \frac{\text{sin}x}{\text{cos}x} \text{cos} x\).
02

Simplify the expression

Now we need to simplify \(\frac{\text{sin}x}{\text{cos}x} \text{cos} x\). The \(\text{cos} x\) terms cancel each other out, so we are left with \(\text{sin} x\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, denoted as \(\tan x\), is a fundamental trigonometric function. It is defined as the ratio of the sine function to the cosine function: \(\tan x = \frac{\text{sin}x}{\text{cos}x}\). This identity is very useful when working with trigonometric expressions because it allows you to express the tangent function in terms of sine and cosine. For example, in the exercise given, we used this definition to transform \(\tan x \text{cos} x\) into \(\frac{\text{sin}x}{\text{cos}x} \text{cos} x\). This makes it easier to simplify further. Understanding this relationship is key to mastering trigonometric identities.
Sine Function
The sine function, represented as \(\text{sin} x\), is one of the basic trigonometric functions. It gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The sine of an angle is crucial in various trigonometric identities and simplifies many expressions. In our exercise, we ended up with the sine function after canceling out the \(\text{cos} x\) terms. This is because when you multiply or divide trigonometric functions, you're often left with a simpler form, which in this case was \(\text{sin} x\).
Cosine Function
The cosine function, denoted as \(\text{cos} x\), is another foundational trigonometric function. It represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Cosine interacts with sine and tangent in various trigonometric identities. In our given problem, we used cosine to simplify the expression: \(\tan x \text{cos} x\). By substituting the definition of tangent, \(\frac{\text{sin}x}{\text{cos}x}\), and then multiplying by \(\text{cos} x\), the \(\text{cos} x\) terms canceled out, leaving us with \(\text{sin} x\). This step highlights the interdependent nature of trigonometric functions.
Simplifying Expressions
Simplifying expressions in trigonometry often involves using fundamental identities to rewrite terms in a more manageable form. In the exercise, we started with \(\tan x \text{cos} x\). By substituting \(\tan x\) with \(\frac{\text{sin}x}{\text{cos}x}\), we transformed the expression into \(\frac{\text{sin}x}{\text{cos}x} \text{cos} x\). Canceling out the \(\text{cos} x\) terms simplified it further to just \(\text{sin} x\).
Simplifying expressions often makes calculations easier and cleaner, and understanding how to use these identities effectively is a valuable skill in trigonometry. Always try to recognize patterns and substitutions that can simplify your work.

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

Verify that each equation is an identity. $$\cos 2 y=\frac{1-\tan ^{2} y}{1+\tan ^{2} y}$$

Verify that each equation is an identity. $$\frac{1-\cos ^{2}\left(\frac{x}{2}\right)}{1-\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1-\cos x}{1+\cos x}$$

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$2 \sin ^{2}\left(\frac{\theta}{2}\right)=\cos \theta$$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (\alpha / 2)=-1 / 3 \text { and } 7 \pi / 4<\alpha / 2<2 \pi$$

Use identities to simplify each expression. Do not use a calculator. $$2 \cos ^{2}\left(22.5^{\circ}\right)-1$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.