Mathematics
Verifying Trigonometric Identities
Verifying trigonometric identities involves proving that one side of a trigonometric equation is equal to the other side. This is typically done by manipulating the expressions using trigonometric identities, algebraic techniques, and properties of trigonometric functions. The goal is to show that both sides of the equation are equivalent, thus verifying the identity.
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9 Key excerpts on "Verifying Trigonometric Identities"
eBook - PDF- 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.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Remember that a conditional equation is an equation that is true for only some of the values in the domain of the variable. For example, the conditional equation sin x = 0 Conditional equation is true only for x = nπ where n is an integer. 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. 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. For example, in Exercise 51 on page 521, trigonometric identities can help you simplify an equation that models the length of a shadow cast by a gnomon (a device used to tell time). 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, or create a denominator with a single term. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Remember that a conditional equation is an equation that is true for only some of the values in the domain of the variable. For example, the conditional equation sin x = 0 Conditional equation is true only for x = nπ where n is an integer. 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. 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. For example, in Exercise 51 on page 361, trigonometric identities can help you simplify an equation that models the length of a shadow cast by a gnomon (a device used to tell time). 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, or create a denominator with a single term. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want.
No longer available |Learn more- Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
Also called even-odd identities. trigonometric identity A statement expressing the equality of two trigonometric expressions. verification of identities A process of algebraic manipulation and substitution through which one can establish the equality of two trigonometric expressions. 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.
eBook - PDFPrecalculus
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
■ Apply various strategies for verifying identities. Verifying Identities 506 Chapter 6 Trigonometric Identities and Equations The objective of this section is to determine whether certain trigonometric equations are identities. In general, there are two basic approaches to verifying identities, as shown below. General Approaches for Verifying Identities ■ Approach 1: One Side of the Equation Focus on the side of the equation with the more complicated expression and transform it until it is identical to the expression on the other side of the equation. ■ Approach 2: Both Sides of the Equation Transform the expressions on both sides of the equation until the resulting expressions on both sides of the equation are identical. As you work to verify identities, you will always use one of these two methods. In addition to these two general approaches, we will also use some specific strategies to verify identities. Strategy 1 Write in Terms of Sine and/or Cosine Write a given expression in terms of sine and/or cosine, and then simplify the resulting expression if necessary. Verify an Identity by Reducing to Sine and Cosine Functions Verify the following identity. (csc x tan x )(sec x ) = sec 2 x Solution Begin with the left-hand side of the equation, because that expression looks more complex: (csc x tan x ) (sec x ) = ( 1 ____ sin x · sin x _____ cos x ) ( 1 _____ cos x ) Use csc x = 1 ____ sin x , tan x = sin x ____ cos x , and sec x = 1 _____ cos x = ( 1 _____ cos x )( 1 _____ cos x ) Simplify inside parentheses = 1 _____ cos 2 x = ( 1 _____ cos x ) 2 = sec 2 x Use 1 ____ cos x = sec x Thus, we have verified that the left side expression, (csc x tan x )(sec x ), can be transformed to the expression on the right, sec 2 x . Check It Out 2 Verify the following identity: (sec x cot x )(sin 2 x ) = sin x Example 2 6.1 Verifying Identities 507 Verify an Identity by Reducing to Sine and Cosine Functions Verify the following identity.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
◆◆◆ Example 8: Simplify Solution: We enter the expression and notice that it does not simplify by default, so we use tCollect. We get ◆◆◆ Since a trigonometric expression can take so many different forms, the calculator may not simplify it in a useful way. Proving a Trigonometric Identity ■ Exploration: Try this. In the same window, graph and using a heavier line for y 2 . What do you see? ■ In some problems we will be asked to manipulate a trigonometric expression so that it matches another expression. For example, you may be asked to verify or prove that the expressions in our exploration are equal, that is, This equation is called an identity because it is true for all values of the variable x for which the functions are defined (for example, the tangent is not defined at ). To prove an identity, we manipulate one or both sides until both sides match. For this we use the fundamental identities and basic operations, such as factoring, reducing fractions, and so forth. We work each side separately and do not treat the identity as if it was an equation, for which we could transpose terms and multiply both sides by the same quantity, for example. If one side of the identity is more complicated than the other, it is a good idea to start by simplifying that side. ◆◆◆ Example 9: Prove the following identity. Solution: Let us try expressing each trigonometric function in terms of the sine and cosine. Thus, and The denominator of the fraction is thus equal to 1, and the identity is proved. ◆◆◆ 1 sin x a 1 cos x b a cos x sin x b 1 1 sin x sec x cot x 1 cot x cos x> sin x. sec x 1> cos x 1 sin x sec x cot x 1 x 90° tan x cos x sin x y 2 sin x y 1 tan x cos x cos u 1 sin u tan u 1 cos u cos u 1 sin u tan u TI-89 screen for Example 8. Section 1 ◆ Fundamental Identities 467 ◆◆◆ Example 10: Prove the identity Solution: We write each expression on the left in terms of sines and cosines, and simplify.
eBook - PDFPre-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.
eBook - PDF- 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.
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
Given an equation with two variables, the method consists of writing the equation in such a way that each side of the equation contains only one type of variable. In Exercises 65–68, use the product-to-sum and sum-to-product identities to separate the variables x and y in each equation. 65. sin a x 1 y 2 b sin a x 2 y 2 b 5 1 5 67. sin1 x 1 y 2 5 1 1 sin1 x 2 y 2 66. 1 2 5 sin a x 1 y 2 b cos a x 2 y 2 b 68. 2 1 cos 1 x 1 y 2 5 cos 1 x 2 y 2 546 CHAPTER 6 Analytic Trigonometry SKILLS OBJECTIVES ■ ■ Find exact values of an inverse sine function. ■ ■ Find exact values of an inverse cosine function. ■ ■ Find exact values of an inverse tangent function. ■ ■ Find exact values of the cotangent, cosecant, and secant inverse functions. ■ ■ Use identities to find exact values of trigonometric expressions involving inverse trigonometric functions. CONCEPTUAL OBJECTIVES ■ ■ Understand that the domain of the sine function is restricted to c 2 p 2 , p 2 d in order for the inverse sine function to exist. ■ ■ Understand that the domain of the cosine function is restricted to 3 0, p4 in order for the inverse cosine function to exist. ■ ■ Understand that the domain of the tangent function is restricted to a 2 p 2 , p 2 b in order for the inverse tangent function to exist. ■ ■ Understand that the cotangent, cosecant, and secant inverse functions are not found from the reciprocal of the tangent, sine, and cosine, respectively, but rather from the inverse secant, inverse cosecant, and inverse cotangent identities. ■ ■ Visualize the quadrants in order to find exact values of trigonometric expressions involving inverse trigonometric functions. 6.5 INVERSE TRIGONOMETRIC FUNCTIONS In Section 1.5, we discussed one-to-one functions and inverse functions. Here we present a summary of that section. A function is one-to-one if it passes the horizontal line test: No two x-values map to the same y-value. Notice that the sine function does not pass the horizontal line test.
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