Trigonometric Identities

11 Key excerpts on "Trigonometric Identities"

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  n Chapter 7, we were given an introduction to the six trigonometric functions, including the definitions of those functions and some of their applications. Now we turn our attention to the ways in which the trigonometric functions are related and how expres-sions involving them can be rewritten in terms of the other trigonometric functions or even as functions of other angles. Our ultimate goal is to be able to solve trigonometric equations, but in order to accomplish this, we must first take a detour through the rich topic of trigono-metric identities. An identity is an equation that is true for all values of the independent variable, and a trigonometric identity (in particular) is an equation that is true for all values of an angle. In this chapter, we’ll investigate some of the funda-mental Trigonometric Identities, including those based on the famous Pythagorean Theorem and those based on the reciprocal relationships existing among the trigonometric functions. We will also search for rela-tionships between the trigonometric functions of the sum and difference of two angles as well as functions of a multiple of an angle. In turn, we will then learn to derive new identities based on those that will be intro-duced to us. I Trigonometric Identities Chapter 8 8.1 I NTRODUCTION TO T RIGONOMETRIC I DENTITIES AND T RIGONOMETRIC F UNCTIONS 8.2 M ORE T RIGONOMETRIC I DENTITIES : S UMS AND D IFFERENCES OF A NGLES , D OUBLE A NGLES , H ALF A NGLES , AND THE Q UOTIENT I DENTITIES 8.3 V ERIFICATION OF F URTHER I DENTITIES Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Analytic Trigonometry with Applications

  • Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  4.1 Fundamental Identities and Their Use 4.2 Verifying Trigonometric Identities 4.3 Sum, Difference, and Cofunction Identities 4.4 Double-Angle and Half-Angle Identities 4.5 Product–Sum and Sum–Product Identities Chapter 4 Group Activity: From M sin Bt + N cos Bt to A sin(Bt + C) Chapter 4 Review 4 Identities 227 ✩ Sections marked with a star may be omitted without loss of continuity. ✩ I I n algebra, we use techniques such as factoring and completing the square to rewrite an expression in order to solve an equation or analyze a function. In trigonometry, we use identities to achieve that purpose. In this chapter we explain what an identity is, and we develop a large number of trigonometric iden- tities. You will get practice in using identities to convert trigonometric expressions into equivalent forms and to solve a variety of problems. 4.1 FUNDAMENTAL IDENTITIES AND THEIR USE • Fundamental Identities • Evaluating Trigonometric Functions • Converting to Equivalent Forms ■ ■ Fundamental Identities An equation in one or more variables is said to be an identity if the left side is equal to the right side for all replacements of the variables for which both sides are defined. The equation is an identity, while is not. The latter is called a conditional equation, since it is true only for certain values of x and not for all values for which both sides are defined. To show that is not an identity, that is, to show that it is a conditional equa- tion, it's not necessary to solve the equation; just find one value of x for which both sides are defined but are unequal. Let x = 0, for example. Then the left side equals -6 and the right side equals 0, so the equation is not an identity. Our first encounter with Trigonometric Identities was in Section 2.5, where we established several fundamental identities. We restate and name these identities in the box for convenient reference. These fundamental identities will be used often in the work that follows.

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  Trigonometry

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  When you are finding the values of the variable for which the equation is true, you are solving the equation. On the other hand, an equation that is true for all real values in the domain of the variable is an identity. For example, the familiar equation sin 2 x = 1 - cos 2 x Identity is true for all real numbers x. So, it is an identity. Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying Trigonometric Identities, the process is best learned through practice. Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. When the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even making an attempt that leads to a dead end can provide insight. Verifying Trigonometric Identities is a useful process when you need to convert a trigonometric expression into a form that is more useful algebraically. When you verify an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication. Trigonometric Identities enable you to rewrite trigonometric equations that model real-life situations.

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  Precalculus with Limits

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  3. Develop additional Trigonometric Identities. 4. Solve trigonometric equations. Pythagorean identities are sometimes used in radical form such as sin u = ±√1 - cos 2 u or tan u = ±√sec 2 u - 1 where the sign depends on the choice of u. Fundamental Trigonometric Identities are useful in simplifying trigonometric expressions. For example, in Exercise 59 on page 354, you will use Trigonometric Identities to simplify an expression for the coefficient of friction. Fundamental Trigonometric Identities Reciprocal Identities sin u = 1 csc u cos u = 1 sec u tan u = 1 cot u csc u = 1 sin u sec u = 1 cos u cot u = 1 tan u Quotient Identities tan u = sin u cos u cot u = cos u sin u Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u Cofunction Identities sin ( π 2 - u ) = cos u cos ( π 2 - u ) = sin u tan ( π 2 - u ) = cot u cot ( π 2 - u ) = tan u sec ( π 2 - u ) = csc u csc ( π 2 - u ) = sec u EvenOdd Identities sin(-u) = -sin u cos(-u) = cos u tan(-u) = -tan u csc(-u) = -csc u sec(-u) = sec u cot(-u) = -cot u © f11photo/Shutterstock.com ALGEBRA HELP You should learn the fundamental Trigonometric Identities well, because you will use them frequently in trigonometry and they will also appear in calculus. Note that u can be an angle, a real number, or a variable. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  In contrast, the equation 2( − 1) =  is not an identity since the only value of  that makes it true is  = 2. Trigonometric Identities allow us to rewrite expressions so we can easily identify the behavior of trigonometric models. Using Identities to Simplify Expressions Trigonometric Identities can help us simplify expressions. Example 1 Use the identities sin(−) = − sin  and cos(∕2 − ) = sin  to rewrite the following as a sinusoidal function and give its amplitude, midline, and period:  = 2 sin  − 3 sin(−) + 4 cos (  2 −  ) . Solution We have  = 2 sin  − 3 sin(−)  − sin  +4 cos (  2 −  )  sin  320 Chapter 9 Trigonometric Identities, POLAR COORDINATES, AND COMPLEX NUMBERS = 2 sin  − 3 (− sin ) + 4 sin  = 2 sin  + 3 sin  + 4 sin  = 9 sin . So this is a sinusoidal function with amplitude  = 9, midline  = 0, and period  = 2. Example 2 Simplify the expression (2 cos  + 3 sin ) (3 cos  + 2 sin ) − 13 sin  cos . Solution To make the calculations easier, let  = cos  and  = sin . Our expression becomes (2 + 3)(3 + 2) − 13 = 6 2 + 4 + 9 + 6 2 − 13 multiply out = 6 2 + 6 2 + 13 − 13 regroup = 6 (  2 +  2 ) simplify and factor = 6 ( cos 2  + sin 2  ) because  = cos ,  = sin  = 6. because cos 2  + sin 2  = 1 We see that this complicated expression equals 6. Using Identities to Evaluate Expressions If we know which quadrant an angle is in and we know the value of any one of its trigonometric functions, we can evaluate others using the Pythagorean identity. Example 3 Suppose that cos  = 2∕3 and 3∕2 ≤  ≤ 2. Find sin  and tan . Solution Substituting cos  = 2∕3 into the identity cos 2  + sin 2  = 1, we get: ( 2 3 ) 2 + sin 2  = 1 4 9 + sin 2  = 1 sin 2  = 1 − 4 9 = 5 9 sin  = ±  5 9 = ± √ 5 3 . Since  is in the fourth quadrant, sin  is negative, so sin  = − √ 5∕3. For tan , we have tan  = sin  cos  = − √ 5∕3 2∕3 = − √ 5 2 .

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  3. Develop additional Trigonometric Identities. 4. Solve trigonometric equations. Pythagorean identities are sometimes used in radical form such as sin u = ±√1 - cos 2 u or tan u = ±√sec 2 u - 1 where the sign depends on the choice of u. Fundamental Trigonometric Identities are useful in simplifying trigonometric expressions. For example, in Exercise 59 on page 514, you will use Trigonometric Identities to simplify an expression for the coefficient of friction. Fundamental Trigonometric Identities Reciprocal Identities sin u = 1 csc u cos u = 1 sec u tan u = 1 cot u csc u = 1 sin u sec u = 1 cos u cot u = 1 tan u Quotient Identities tan u = sin u cos u cot u = cos u sin u Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u Cofunction Identities sin ( π 2 - u ) = cos u cos ( π 2 - u ) = sin u tan ( π 2 - u ) = cot u cot ( π 2 - u ) = tan u sec ( π 2 - u ) = csc u csc ( π 2 - u ) = sec u EvenOdd Identities sin(-u) = -sin u cos(-u) = cos u tan(-u) = -tan u csc(-u) = -csc u sec(-u) = sec u cot(-u) = -cot u © f11photo/Shutterstock.com ALGEBRA HELP You should learn the fundamental Trigonometric Identities well, because you will use them frequently in trigonometry and they will also appear in calculus. Note that u can be an angle, a real number, or a variable. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  461 16 Trigonometric Identities and Equations ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to • Write a trigonometric expression in terms of the sine and cosine. • Simplify a trigonometric expression using the fundamental identities. • Prove Trigonometric Identities using the fundamental identities. • Simplify expressions or prove identities using the sum or difference formulas, the double-angle formulas, or the half-angle formulas. • Evaluate trigonometric expressions. • Solve trigonometric equations. ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ In mathematics we usually try to simplify expressions as much as possible. In ear- lier chapters, we simplified algebraic expressions of all sorts. In this, our final chap- ter on trigonometry, we will simplify trigonometric expressions. For example, an expression such as tan x cos x simplifies to sin x. For this we need to know how various trigonometric functions are related. We will start with the simplest (and most useful) fundamental identities. These identi- ties are equations relating one trigonometric expression to another. Using them, we can replace one expression with another that will lead to a simpler result. We then proceed to trigonometric expressions containing sums and differences of two an- gles, double angles, and half angles. This is followed by a short section on evaluating trigonometric expressions and another on solving trigonometric equations. We approximately found roots of a trigonometric equation by calculator in the preceding chapter, and here we learn how to do an exact solution. For example, we know how to find the vertical and horizontal displacements of a projectile, Fig. 16–1, given the initial velocity and the launch angle But how would we solve for given the other quantities? We will learn how in this chapter. u, u. 0  x y FIGURE 16–1

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  18 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Simplify trigonometric expressions using fundamental identities. • Expand or simplify trigonometric expressions containing the sum or difference of two angles. • Simplify trigonometric expressions containing double angles. • Simplify trigonometric expressions containing half angles. • Solve trigonometric equations. • Evaluate inverse trigonometric functions. You have used trigonometry in solving vectors and geometry problems, and in working with machinery and navigation. Essentially, anywhere you could model a situation with right tri- angles you could use trigonometry (and in some cases, using the laws of sines and cosines, you didn’t even need to break the situation down into right triangles). As you study more realistic conditions and things get a little more complex, you’ll start to see more trigonometric functions in equations. In technical mathematics, we have found some very useful identities that will help you manipulate an equation to find a solution to a problem. An identity is something that is true for a wide range of variables. Trigonometric Identities and Equations 18–1 Fundamental Identities Reciprocal Relations We have already encountered the reciprocal relations in Sec. 7–2, and we repeat them here. Reciprocal Relations θ θ θ θ θ θ = = = sin 1 csc or csc 1 sin or sin csc 1 152a θ θ θ θ θ θ = = = cos 1 sec or sec 1 cos or cos sec 1 152b θ θ θ θ θ θ = = = tan 1 cot or cot 1 tan or tan cot 1 152c ◆◆◆ Example 1: Simplify cos sec 2 θ θ . Solution: Using Eq. 152b gives us cos sec cos (cos ) cos 2 2 3 θ θ θ θ θ = = Quotient Relations Figure 18-1 shows an angle θ in standard position, as when we first defined the trigonometric functions in Sec.

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  T he number of daylight hours varies during different times of the year and can be modeled by a trigonometric function. You can solve a related trigonometric equation to calculate the time of year when you have a specified number of daylight hours at a particular location. See Exercises 98 and 99 in Section 6.4. In this chapter, we will study relations between the different trigonometric functions by establishing identities and using them to solve equations. Chapter 6 Outline – 1 1 y x – 2 2 – – 3 p 2 3 p 2 p p 2 1 y = y = cos x 2 p p p 6.1 Verifying Identities 6.2 Sum and Difference Identities 6.3 Multiple-Angle Identities; Sum and Product Identities 6.4 Trigonometric Equations Trigonometric Identities and Equations 6.1 6.1 Verifying Identities 505 When working with trigonometric functions, it is useful to transform expressions containing these functions into an equivalent form. Consider the equation cos 2 x = 1 − sin 2 x , where cos 2 x is written in terms of sin 2 x . This is an example of an identity, because it holds true for all val-ues of x . The process of demonstrating that an equation holds true for all values of a variable for which the terms of the equation are defined is called verifying an identity. The process is also referred to as establishing or proving an identity. Basic Identities Table 1 summarizes some familiar Trigonometric Identities. Identities as Ratios tan x = sin x ____ cos x cot x = cos x ____ sin x Reciprocal Identities sec x = 1 ____ cos x csc x = 1 ____ sin x cot x = 1 ____ tan x Pythagorean Identities cos 2 x + sin 2 x = 1 1 + tan 2 x = sec 2 x 1 + cot 2 x = csc 2 x Negative-Angle Identities cos ( − x ) = cos x sin ( − x ) = − sin x tan ( − x ) = − tan x Table 1 To show that an equation is an identity, we must show that the equations are true for all values of the variable for which the expressions are defined.

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  Pre-Calculus All-in-One For Dummies

  Book + Chapter Quizzes Online

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  This information is truly helpful when you get to calculus, which takes these calculations to another level (a level at which you integrate and differentiate by using these identities). Chapter 13 IN THIS CHAPTER » Applying the sum and difference formulas of trig functions » Utilizing double-angle formulas » Cutting angles in two with half- angle formulas » Changing from products to sums and back » Tossing aside exponents with power-reducing formulas 298 UNIT 4 Identities and Special Triangles Finding Trig Functions of Sums and Differences Long ago, some fantastic mathematicians found identities that hold true when adding and sub- tracting angle measures from special triangles (30 60 90 right triangles and 45 45 90 right triangles; see Chapter 9). The focus is to find a way to rewrite an angle as a sum or differ- ence of two “convenient” angles. Those mathematicians were curious; they could find the trig values for the special triangles but wanted to know how to deal with other angles that aren’t part of the special triangles on the unit circle. They could solve problems with multiples of 30 and 45 , but they knew nothing about all the other angles that couldn’t be formed that way! Constructing these angles was simple; however, evaluating trig functions for them proved to be a bit more difficult. So they put their collective minds together and discovered the sum and difference identities discussed in this section. Their only problem was that they still couldn’t find the trig values of many other angles using the sum a b and difference a b formulas. This section takes the information covered in earlier chapters, such as calculating trig values of special angles, to the next level. You are advanced to identities that allow you to find trig values of angles that are multiples of 15 . Note: You’ll never be asked to find the sine of 87 , for example, using trig identities.

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  CONCEPTUAL OBJECTIVES • Understand that the sum and difference identities are used to derive product-to-sum identities. • Understand that the product-to-sum identities are used to derive the sum-to-product identities. Often in calculus it is helpful to write products of trigonometric functions as sums of other trigonometric functions and vice versa. In this section, we discuss the product-to-sum identities, which convert products of trigonometric functions to sums of trigonometric functions, and sum-to-product identities, which convert sums of trigonometric functions to products of trigonometric functions. 6.4.1 Product-to-Sum Identities 6.4.1 Skill Express products of trigonometric functions as sums of trigonometric functions. 6.4.1 Conceptual Understand that the sum and difference identities are used to derive product-to-sum identities. The product-to-sum identities are derived from the sum and difference identities. Words Math Write the identity for the cosine of a sum. cos A cos B − sin A sin B = cos(A + B) Write the identity for the cosine of a difference. cos A cos B + sin A sin B = cos(A − B) Add the two identities. 2 cos A cos B = cos(A + B) + cos(A − B) Divide both sides by 2. cos A cos B = 1 _ 2 [ cos(A + B) + cos(A − B)] 548 CHAPTER 6 Analytic Trigonometry Subtract the sum identity from the cos A cos B + sin A sin B = cos(A − B) difference identity. −cos A cos B + sin A sin B = −cos(A + B) 2 sin A sin B = cos(A − B) − cos(A + B) Divide both sides by 2. sin A sin B = 1 _ 2 [ cos(A − B) − cos(A + B)] Write the identity for the sine of a sum. sin A cos B + cos A sin B = sin(A + B) Write the identity for the sine of a difference. sin A cos B − cos A sin B = sin(A − B) Add the two identities. 2 sin A cos B = sin(A + B) + sin(A − B) Divide both sides by 2. sin A cos B = 1 _ 2 [ sin(A + B) + sin(A − B)] Product-to-Sum Identities 1. cos A cos B = 1 _ 2 [cos(A + B) + cos(A − B)] 2. sin A sin B = 1 _ 2 [cos(A − B) − cos (A + B)] 3.

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