/*! 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 50 Verify each identity. $$\frac{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 50

Verify each identity. $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}=\frac{2 \sec t+1}{\sec t}$$

Short Answer

Expert verified
After simplifying both sides of the equation, it's shown that the left-hand side equals the right-hand side. Thus, the given equation is indeed an identity.

Step by step solution

01

Simplify the Left-Hand Side

The left-hand side (LHS) is a quadratic expression divided by a linear expression in terms of \( \cos t \). The quadratic can be factored as \( (\cos t + 2)^2 \). So, the LHS simplifies to \( \cos t + 2 \) .
02

Simplify the Right-Hand Side

The right-hand side (RHS) is a linear expression divided by a linear expression in terms of \( \sec t \). We should recall that \( \sec t = 1/ \cos t \). If we substitute \( \sec t \) with \( 1/ \cos t \), the RHS becomes \( \frac{2/ \cos t + 1}{1/ \cos t}\). Simplifying this, we get \( \cos t + 2 \).
03

Compare LHS and RHS

Comparing both sides, we see that both sides are equal, hence the given equation is an identity.

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.

Trigonometric Identities
Trigonometric identities are fundamental to understanding and solving problems in trigonometry. They are equations involving trigonometric functions like sine, cosine, tangent, etc., that hold true for all values within their domains. These identities are useful for simplifying complex expressions and proving other mathematical properties. When verifying an identity, the goal is to manipulate one side of the equation (or both sides independently) to match the other. The strategy often involves algebraic manipulation such as factoring, expanding, adding fractions, and using known trigonometric identities like the Pythagorean identities or reciprocal relationships.
Cosine Function
The cosine function, denoted as \( \cos \) and one of the primary trigonometric functions, relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. In the unit circle, it represents the x-coordinate of a point. This function is even, meaning \( \cos(-t) = \cos(t) \), and has a range of \( [-1,1] \). Understanding how to simplify expressions with cosine, including those raised to a power, is crucial when verifying trigonometric identities. It's also essential to know its reciprocal function, the secant function, to simplify complex trigonometric equations.
Secant Function
The secant function is the reciprocal of the cosine function, represented as \( \sec t = \frac{1}{\cos t} \). It's an important trigonometric function used to transform trigonometric expressions and verify identities. The secant function can simplify solving equations by replacing a complicated expression with a simpler equivalent. When \( \cos t \) appears in the denominator, using \( \sec t \) aids in simplifying and can lead to more straightforward expressions or solutions.
Factoring Quadratic Expressions
Factoring is a vital algebraic process, especially when working with quadratic expressions such as \( ax^2 + bx + c \). To factor a quadratic expression, one seeks to write it as the product of two binomial expressions. This is particularly helpful in trigonometric identities where the quadratic might involve trigonometric functions. Factoring can simplify the expression, making it easier to identify and verify the trigonometric identity. A common technique used is to find two numbers that multiply to give the product of the coefficient of \( x^2 \) and the constant term, and add up to the coefficient of the \( x \) term.
Simplifying Expressions
Simplifying expressions is a key aspect of algebra and trigonometry. It involves combining like terms, using basic arithmetic, applying algebraic rules, and implementing trigonometric identities to rewrite expressions in a more manageable form. In the context of trigonometric identities, simplifying the expressions on each side of the equation allows for easier comparison and verification. The ability to simplify intricate expressions is invaluable for solving problems, proving identities, and even in calculus where trigonometric simplifications can help in integration and differentiation.

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

Find the exact value of each expression. Do not use a calculator. $$\cos ^{2}\left(\frac{1}{2} \sin ^{-1} \frac{3}{5}\right)$$

Use this information to solve Exercises \(131-132 .\) The number of hours of daylight in Boston is given by $$ y=3 \sin \left[\frac{2 \pi}{365}(x-79)\right]+12 $$ where \(x\) is the number of days after January 1 Within a year, when does Boston have 10.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day.

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$5 \sin x=2 \cos ^{2} x-4$$

Use this information to solve Exercises \(135-136 .\) When throwing an object, the distance achieved depends on its initial velocity, \(\bar{v}_{0}\) and the angle above the horizontal at which the object is thrown, \(\theta\). The distance, \(d,\) in feet, that describes the range covered is given by $$ d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta $$ where \(v_{0}\) is measured in feet per second. You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(v_{0}=90\) feet per second, at what angle of elevation, \(\theta,\) to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?

In Exercises \(121-126,\) solve each equation on the interval \([0,2 \pi)\) $$|\sin x|=\frac{1}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.