Mathematics
Pythagorean Identities
Pythagorean identities are a set of trigonometric equations that relate the three basic trigonometric functions: sine, cosine, and tangent, with the Pythagorean theorem. These identities are used to simplify trigonometric expressions and solve trigonometric equations.
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11 Key excerpts on "Pythagorean Identities"
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)
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. 282 Chapter 8 We include the following remark, prior to moving on to the use of trig-onometric identities: each of those Pythagorean Identities can be rewritten algebraically in another form, and those alternate forms are also considered Pythagorean Identities (but are suppressed from the general list). For instance, the first identity could be restated as sin 2 5 1 2 cos 2 . At times, it is useful to think of the Pythagorean Identities from this alternate perspective, as we’ll see shortly. Identities Used to Simplify Trigonometric Expressions A trigonometric expression is any mathematical statement involving one or more of the trigonometric functions. For instance, cos 2 3 sin 2 tan is a trigonometric expression. If we examine such statements, it is possible that one of the identities we have presented can be used to rewrite the expression in a different form. The form may be aesthetically more pleasing, or it may be that later, when we attempt to prove further trigonometric identities, the alterna-tive forms will be easier to work with. Why that would be the case is absolutely unclear at the moment, but we will develop an appreciation for that statement in the near future. Consider, as an early illustration, the following expression: cos 4 2 sin 4 Note that this is not an equation, since there is no equal sign present! It is a common mistake to use the terms “equation” and “expression” interchangeably, but we really should be precise here.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
125 through by or 126 Finally, we get a third Pythagorean relation by dividing Eq. 125 through by or 127 ◆◆◆ Example 3: Simplify Solution: By the Pythagorean relations, ◆◆◆ Simplifying a Trigonometric Expression One use of the trigonometric identities is the simplification of expressions, as in the preceding examples. We now give a few more examples. ◆◆◆ Example 4: Simplify Solution: We start by replacing But so Finally, since ◆◆◆ (cot 2 u 1)(sec 2 u 1) sec 2 u 1 cos u sec u, 1 cos 2 u csc 2 u tan 2 u 1 sin 2 u # sin 2 u cos 2 u csc 2 u 1 sin 2 u and tan 2 u sin 2 u cos 2 u , (cot 2 u 1)(sec 2 u 1) csc 2 u tan 2 u (cot 2 u 1) by csc 2 u and (sec 2 u 1) by tan 2 u (cot 2 u 1)(sec 2 u 1). 1 (1) (1) 1 (sin 2 u cos 2 u) (tan 2 u sec 2 u) (cot 2 u csc 2 u) sin 2 u csc 2 u tan 2 u cot 2 u cos 2 u sec 2 u sin 2 u csc 2 u tan 2 u cot 2 u cos 2 u sec 2 u 1 cot 2 u csc 2 u Pythagorean Relation sin 2 u sin 2 u cos 2 u sin 2 u 1 sin 2 u sin 2 u cos 2 u 1 sin 2 u. 1 tan 2 u sec 2 u Pythagorean Relation sin 2 u cos 2 u cos 2 u cos 2 u 1 cos 2 u sin 2 u cos 2 u 1 cos 2 u. Section 1 ◆ Fundamental Identities 465 ◆◆◆ Example 5: Simplify Solution: Factoring the difference of two squares in the numerator gives ◆◆◆ ◆◆◆ Example 6: Simplify Solution: Combining the two fractions over a common denominator, we have Replacing with and collecting terms gives Factoring the numerator, Finally, since we get ◆◆◆ Simplifying a Trigonometric Expression by Calculator Some calculators such as the TI-89 can simplify a trigonometric expression. They may even simplify them automatically, as in the following example. ◆◆◆ Example 7: Simplify by calculator, Solution: We simply enter the expression and press . The simplified ex- pression tan x appears in the display.
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
And in some instances, you’re faced with such a conglomeration of functions that figuring out what’s going on is darn near impos- sible. Other times, the different terms have different powers of the same function. In such cases, simplifying matters either by changing everything to sines and cosines or by factoring out some function may be your best bet. THE MUSIC OF THE SPHERES Pythagoras is best known for his theorem, which defines the relationships among the lengths of a right triangle’s sides, but his second most well-known contribution to humanity is his discovery of the mathematical basis of the musical scale. He found that a connection exists between musical harmony — the stuff that sounds good — and whole numbers. If you pluck a taut string, listen to the note, and then pluck a string twice as long and equally taut, you hear a note one octave below the first note. You can also go down the scale by increasing the length of the taut string in smaller increments. Pythagoras believed that whole-number relationships represent all harmony, all beauty, and all nature. He extended this theory to the orbits of the planets and believed that as the planets move through space, they must give off a heavenly whole-number har- mony. Hence the term the music of the spheres (in one of my favorite songs from Les Misérables). 206 PART 3 Identities Changing to sines and cosines In this first example, you can use either reciprocal or ratio identities, depending on which side you’re going to work on, to change everything to sines and cosines. Here’s how I’d solve the identity tan cot csc sec : 1. Going with the guideline to work on the side with the greatest number of terms, replace the two terms on the left by using ratio identities. sin cos cos sin csc sec 2. To get a common denominator, multiply both terms on the left by fractions equal to 1 (by using the other term’s denominator).
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
An approximation of the identity in the next example was used by the Indian Mathematician Aryabhata in about 500 AD to construct a table of sines. EXAMPLE 4.12.3. Prove that α α α α α α ( ) ( ) ( ) ( ) ( ) ( ) ( ) + - = - - - n n n n n sin 1 sin sin sin 1 4 sin sin 2 . 2 Trigonometry • 111 Answer: n n n n n n n n n n n n n n n LHS sin(( 1) ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) sin( ) cos( ) 1 cos( ) sin( ) RHS sin( ) (sin( ) cos( ) cos( ) sin( )) 4 sin( ) 1 cos( ) 2 sin( )(1 cos( ) 2(1 cos( ))) cos( ) sin( ) sin( ) cos( ) 1 cos( ) sin( ) α α α α α α α α α α α α α α α α α α α α α α α α α α α ( ) ( ) = + - = + - = - + = - - - - = - - - + = - + The LHS equals the RHS, so the identity is proved. 4.13 SOLVING TRIANGLES Any triangle has six parts: three sides and three angles. If any three parts of a triangle are given (with at least one of these being the length of a side), then in most cases we can calculate unambiguously what the other parts are by means of the formulas that are derived in this section. In particular, if the triangle is a right triangle, then one angle (the right angle) is automatically given, and if any two other parts are given (with at least one of these being the length of a side), then the trigonometric ratios automatically determine what the other parts are. 4.13.1 Right Triangles The connection between right triangles and trigonometric ratios has already been indicated a few times in this chapter. Any right triangle can be oriented in the Cartesian plane, so that either of its acute angles coincides with an angle measured in a counterclockwise direction from the x-axis; that is, one vertex is situated at the origin, one side (not the hypotenuse) is aligned with the positive x-axis, and the hypotenuse radiates from the origin to another vertex of the right triangle in the first quadrant.
eBook - PDF- 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.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
22 The well-known basic identity satisfied by the sinusoidal functions sin θ and cos θ is cos 2 θ + sin 2 θ = 1 . (1.56) For sin θ and cos θ defined geometrically this is an immediate consequence of the theorem due to Pythagoras. If they have been defined algebraically by means of series then the result from Appendix B is needed as a link to the Pythagorean justification; a more direct proof is available using Euler’s equation (Chapter 5 ). Other standard single-angle formulae derived from ( 1.56 ) by dividing through by vari-ous powers of sin θ and cos θ are 23 1 + tan 2 θ = sec 2 θ, (1.57) cot 2 θ + 1 = cosec 2 θ. (1.58) 1.5.1 Compound-angle identities The basis for building expressions for the sinusoidal functions of compound angles are those for the sum and difference of just two angles, since all other cases can be built up from these, in principle. Later we will see that a study of complex numbers can provide a more efficient approach in some cases. To prove the basic formulae for the sine and cosine of a compound angle A + B in terms of the sines and cosines of A and B , we consider the construction shown in Figure 1.3 . It shows two sets of axes, Oxy and Ox y , with a common origin O , but rotated with respect to each other through an angle A . The point P lies on the unit circle centred on the common origin and has coordinates cos( A + B ) , sin( A + B ) with respect to the axes Oxy and coordinates cos B, sin B with respect to the axes Ox y . Parallels to the axes Oxy (dotted lines) and Ox y (broken lines) have been drawn through P . Further parallels ( MR and RN ) to the Ox y axes have been drawn through R , the point (0 , sin( A + B )) in the Oxy system. That all the angles marked with the symbol • are equal to A follows from the simple geometry of right-angled triangles and crossing lines.
eBook - PDF- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
We can find the latter two val- ues using Pythagorean Identities or, equivalently, by using reference triangles and the Pythagorean theorem. From Figure 3, From Figure 4, Using the sum identity, ■ Matched Problem 4 Find the exact value of given x in quadrant III, and y in quadrant IV. ■ EXAMPLE 5 Verifying an Identity Verify the identity: cot y - cot x = sin(x - y) sin x sin y cos y = 15 > 3, sin x = - 2 3 , sin(x - y), = a - 4 5 b a 4 5 b - a 3 5 b a 3 5 b = - 25 25 = - 1 cos(x + y) = cos x cos y - sin x sin y sin y = 3 5 b = 25 2 - 4 2 = 3 cos x = - 4 5 a = - 25 2 - 3 2 = - 4 cos(x + y) = cos x cos y - sin x sin y cos y = 4 5 , sin x = 3 5 cos(x + y), = 13 + 1 2 12 = 1 12 # 13 2 + 1 12 # 1 2 252 4 IDENTITIES a a 3 5 x b (a, 3) FIGURE 3 a b b y 4 5 (4, b) FIGURE 4 Verification We start with the right side because it involves , while the left side involves only x and y: Use a difference identity. Use algebra. Cancel common factors. Use a quotient identity. Left side ■ Matched Problem 5 Verify the identity: ■ 1. sin x 2. 3. 4. 5. EXERCISE 4.3 = cot y + tan x = tan x + cot y = cos x cos y cos x sin y + sin x sin y cos x sin y cos(x - y) cos x sin y = cos x cos y + sin x sin y cos x sin y - 4 15 9 2 + 13 2 2 2p 2p y 1 = sin(x - p); y 2 = - sin x Answers to Matched Problems tan x + cot y = cos(x - y) cos x sin y = cot y - cot x = cos y sin y - cos x sin x = sin x cos y sin x sin y - cos x sin y sin x sin y = sin x cos y - cos x sin y sin x sin y Right side = sin(x - y) sin x sin y x - y 4.3 Sum, Difference, and Cofunction Identities 253 1. How can you show that an equation in two variables, x and y, is not an identity? 2. Explain what it means for an equation in two variables, x and y, to be an identity. 3. If the left and right sides of an equation in two variables are equal whenever x = y, is the equation an identity? Explain. 4. If the left and right sides of an equation in two variables are equal whenever x = -y, is the equation an identity? Explain.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
The identities developed in this chapter are useful in such applications as musical sound where they allow the determination of the “beat” frequency. In calculus, these identities will simplify the integration and differentiation processes. 7.3.1 Sum and Difference Identities for the Cosine Function 7.3.1 Skill Find exact values for the cosine function using sum and difference identities. 7.3.1 Conceptual Understand that sums or differences of trigonometric functions can be written as a single cosine expression. Derivation of the Sum and Difference Identities for the Cosine Function Recall from Section 6.7 that the unit circle approach for defining trigonometric functions gave the relationship between the coordinates along the unit circle and the sine and cosine functions. Specifically, the x-coordinate corresponds to the value of the cosine function, and the y-coordinate corresponds to the value of the sine function for a given angle θ and the point (x, y) where the terminal side of the angle θ intersects the unit circle. x y (1, 0) (0, 1) (cosθ, sinθ) r = 1 θ (0, –1) (–1, 0) Let us now draw the unit circle with two angles, and , realizing that the two terminal sides of these angles form a third angle, − . x y (1, 0) (0, 1) P 1 P 2 α – β (0, –1) (–1, 0) α β Distance Formula Sum and Difference Identities for Cosine Sum and Difference Identities for Tangent Sum and Difference Identities for Sine Cofunction Identities 7.3 Sum and Difference Identities 689 If we label the points P 1 = (cos , sin ) and P 2 = (cos , sin ), we can then draw a segment P 1 P 2 connecting points P 1 and P 2 . α – β β x y (1, 0) (0, 1) P 2 = (cos β, sin β) P 2 (0, –1) (–1, 0) P 1 = (cos α, sin α) P 1 α If we rotate the angle clockwise so that the central angle − is in standard position, then the two points where the initial and terminal sides intersect the unit circle are P 4 = (1, 0) and P 3 = (cos( − ), sin( − )), respectively.
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 - 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)
- 2021(Publication Date)
- Wiley(Publisher)
5.5 Product-to-Sum and Sum- to-Product Identities SKILLS OBJECTIVES • Express products of trigonometric functions as sums of trigonometric functions. • Express sums of trigonometric functions as products of trigonometric functions. 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. 5.5.1 Product-to-Sum Identities 5.5.1 Skill Express products of trigonometric functions as sums of trigonometric functions. 5.5.1 Conceptual Understand that the sum and difference identities are used to derive product-to-sum identities. 278 CHAPTER 5 Trigonometric 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)] 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.
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