Mathematics
Law of Cosines in Algebra
The Law of Cosines in algebra is a formula used to find the length of a side or measure of an angle in a triangle when the lengths of the other two sides and the included angle are known. It is an extension of the Pythagorean theorem and is particularly useful for solving non-right-angled triangles.
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12 Key excerpts on "Law of Cosines in Algebra"
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
section 10.4 The Law of Sines and the Law of Cosines 631 The Law of Cosines The law of sines is a wonderful tool for finding the lengths of all three sides of a triangle when we know two of the angles of the triangle (which means that we know all three angles) and the length of at least one side of the triangle. Also, if we know the lengths of two sides of a triangle and one of the angles other than the angle between those two sides, then the law of sines allows us to find the other angles and the length of the other side, although it may produce two possible choices rather than a unique solution. However, the law of sines is of no use if we know the lengths of all three sides of a triangle and want to find the angles of the triangle. Similarly, the law of sines cannot help us if the only information we know about a triangle is the length of two sides and the angle between those sides. Fortunately the law of cosines, our next topic, provides the necessary tools for these tasks. As we will see, the law of cosines is a general- ization to all triangles of the Pythagorean Theorem, which ap- plies only to right tri- angles. Consider a triangle with sides of lengths a, b, and c and an angle of C opposite the side of length c, as shown here. C a b c t r h Drop a perpendicular line segment from the vertex opposite the side of length b to the side of length b, as shown above. The length of this line segment is the height of the triangle; label it h. The endpoint of this line segment of length h divides the side of the triangle of length b into two smaller line segments, which we have labeled r and t above. The line segment of length h shown above divides the original larger triangle into two smaller right triangles. Looking at the right triangle on the right, we see that sin C = h a . Thus h = a sin C. Furthermore, looking at the same right triangle, we see that cos C = t a . Thus t = a cos C. The figure above also shows that r = b - t .
eBook - PDFPrecalculus
A Prelude to Calculus
- Sheldon Axler(Author)
- 2016(Publication Date)
- Wiley(Publisher)
The law of sines and law of cosines could have been called the sine theorem and cosine theorem. cosines could be used, as discussed below: • Suppose you know only the lengths of all three sides of a triangle. The only possibility in this situation is first to use the law of cosines to find one of the angles. Then, knowing the lengths of all three sides of the triangle and one angle, you could use either the law of cosines or the law of sines to find another angle. However, the law of sines may lead to two choices for the angle rather than a unique choice; thus it is better to use the law of cosines in this situation. • Another case where you could use either law is when you know the length of two sides of a triangle and an angle other than the angle between those two sides. With the notation from the beginning of this section, suppose we know a, c, and C. We could use either the law of sines or the law of cosines to get an equation with only one unknown: sin A a = sin C c or c 2 = a 2 + b 2 - 2ab cos C. The first equation above, where A is the unknown, may lead to two possible choices for A. Similarly, the second equation above, where b is the unknown and we need to use the quadratic formula to solve for b, may lead to two possible choices for b. Thus both laws may give us two choices. The law of sines is probably a bit simpler to apply. The box below summarizes when to use which law. As usual, you will be better This illustration published in 1621 by Swiss mathematician Peter Ryff shows trigonometry being used to measure the height of a column. off understanding how these guidelines arise (you can then always reconstruct them) rather than memorizing them. If you know two angles of a triangle, then finding the third angle is easy because the sum of the angles of a triangle equals π radians or 180 ◦ .
eBook - PDF- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
8.2.1 CONCEPTUAL Understand that the Pythagorean theorem is a special case of the Law of Cosines. 8.2.1 Solving Oblique Triangles In Section 8.1, we learned that to solve oblique triangles means to find all three side lengths and angle measures. At least one side length must be known. We need two additional pieces of information to solve a triangle (combinations of side lengths and/ or angles). We found that there are four cases: ■ ■ Case 1: AAS or ASA (two angles and a side are given) ■ ■ Case 2: SSA (two sides and an angle opposite one of the sides are given) ■ ■ Case 3: SAS (two sides and the angle between them are given) ■ ■ Case 4: SSS (three sides are given) We used the Law of Sines to solve Case 1 and Case 2 triangles. Now, we use the Law of Cosines to solve Case 3 and Case 4 triangles. A B C X Z Y 65. A 5 10, Y 5 40°, and Z 5 72° 66. B 5 42.8, X 5 31.6°, and Y 5 82.2° 67. A 5 22, B 5 17, and X 5 105° 68. B 5 16.5, C 5 9.8, and Z 5 79.2° 69. A 5 25.7, C 5 12.2, and X 5 65° 70. A 5 54.6, B 5 12.9, and Y 5 23° For Exercises 65–70, let A, B, and C be the lengths of the three sides with X, Y, and Z as the corresponding angles. Write a program to solve the given triangle with a calculator. • T E C H N O L O G Y SKILLS OBJECTIVE ■ ■ Solve SAS and the SSS triangles using the Law of Cosines. CONCEPTUAL OBJECTIVE ■ ■ Understand that the Pythagorean theorem is a special case of the Law of Cosines. 8.2 THE LAW OF COSINES WORDS MATH Start with a triangle. Drop a perpendicular line from g with height h. The result is two triangles within the larger triangle. Write the Pythagorean theorem for both right triangles. Triangle 1: x 2 1 h 2 5 b 2 Triangle 2: 1 c 2 x 2 2 1 h 2 5 a 2 Solve for h 2 . Triangle 1: h 2 5 b 2 2 x 2 Triangle 2: h 2 5 a 2 2 1 c 2 x 2 2 Since the segment of length h is shared, set h 2 5 h 2 , for the two triangles. b 2 2 x 2 5 a 2 2 1 c 2 x 2 2 Multiply out the squared binomial on the right.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
65. The Pythagorean theorem is a special case of the Law of Cosines. 66. The Law of Cosines is a special case of the Pythagorean theorem. 67. In a triangle, the length of the shortest side of the triangle is 1 _ 2 the length of the longest side. The other side of the triangle is 3 _ 4 the length of the longest side. What is the size of the largest angle? 68. In a triangle, the length of one side is 1 _ 4 the length of an adjacent side. If the angle between the sides is 60°, how does the length of the third side compare with that of the longer of the other two sides? Challenge 69. Show that cos α _____ a + cos β _____ b + cos γ _____ c = a 2 + b 2 + c 2 __________ 2 abc . Hint: Use the Law of Cosines. 70. Show that a = c cos β + b cos γ. Hint: Use the Law of Cosines. 71. In an isosceles triangle, the longer side is 50% longer than the other two sides. What is the size of the vertex angle? 72. In an isosceles triangle, the longer side is 2 inches longer than the other two sides. If the vertex angle measures 80°, what are the lengths of the sides? Technology For Exercises 73–76, let A, B, and C be the lengths of the three sides with X, Y, and Z as the corresponding angle measures. Write a program to solve the given triangle. A B C X Z Y 73. B = 45, C = 57, and X = 43° 74. B = 24.5, C = 31.6, and X = 81.5° 75. A = 29.8, B = 37.6, and C = 53.2 76. A = 100, B = 170, and C = 250 8.3 The Area of a Triangle 799 8.3 The Area of a Triangle SKILLS OBJECTIVES • Find the area of a triangle in the SAS case. • Find the area of a triangle in the SSS case. CONCEPTUAL OBJECTIVES • Understand how to derive a formula for the area of a triangle (SAS case) using the Law of Sines. • Understand how to derive a formula for the area of a triangle (SSS case) using the Law of Cosines. In Sections 8.1 and 8.2, we used the Law of Sines and the Law of Cosines to solve oblique triangles, which means to find all of the side lengths and angle measures.
- Alan Sultan, Alice F. Artzt(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
Suppose we have one triangle and we only know the measures of two of its angles and one side. Would we be able to use the Law of Cosines to determine information about the other two sides? Well, the answer is, “No.” Since the Law of Cosines requires us to know two sides and one angle, we do not have enough information to use it. To find the missing information about the triangle, we need another law which we will discuss in the next section: the Law of Sines.Student Learning Opportunities- 1* Given that the sides of a triangle are a = 3, b = 5, and c = 7, find all three angles.
- 2* If the sides of a parallelogram are 3 and 4 and the angle between them is 30 degrees, how long is each diagonal?
- 3* A surveyor needs to estimate the distance across a lake from point A to point B . Standing at point C , 4.6 miles from A and 7.3 miles from B , he measures the angle shown in Figure 5.5 to be 80 degrees. Estimate the distance AB .
Figure 5.5
- 4 We will point out in the chapter on trigonometry that cos(180° − x ) = −cos x . Use this fact to prove the Law of Cosines when the altitude AD in Figure 5.1 is outside of the triangle.
- 5 (C) A student asks how you prove the other two versions of the Law of Cosines found in equations (5.9) and (5.10) . How do you do it?
- 6* Your students are intrigued by how the Pythagorean Theorem was used to prove the Law of Cosines. They wonder because of the similar structure of the theorems, if one can go in reverse. That is, can one use the Law of Cosines to prove that if c 2 = a 2 + b 2 holds in a triangle, then the triangle is right? What is your answer and how do you show it?
- 7 (C) A student asks how you can prove that if two angles and any side of one triangle are equal to two angles and the corresponding side of another triangle, the triangles are congruent (i.e., AAS = AAS). How do you prove it?
5.3 The Law of SinesLaunchGive an example (draw it) where two angles and a side of one triangle are equal to two angles and a side of another triangle, but the triangles are not congruent. [Hint: make sure the sides are not corresponding.]
eBook - PDF- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
63. The Pythagorean theorem is a special case of the Law of Cosines. 64. The Law of Cosines is a special case of the Pythagorean theorem. 65. In a triangle, the length of the shortest side of the triangle is 1 2 the length of the longest side. The other side of the triangle is 3 4 the length of the longest side. What is the size of the largest angle? 66. In a triangle, the length of one side is 1 4 the length of an adjacent side. If the angle between the sides is 60°, how does the length of the third side compare with that of the longer of the other two? 67. Show that cos a a 1 cos b b 1 cos g c 5 a 2 1 b 2 1 c 2 2 abc . Hint: Use the Law of Cosines. 68. Show that a 5 c cos b 1 b cos g. Hint: Use the Law of Cosines. 69. In an isosceles triangle, the longer side is 50% longer than the other two sides. What is the size of the vertex angle? 70. In an isosceles triangle, the longer side is 2 inches longer than the other two sides. If the vertex angle measures 80°, what are the lengths of the sides? • C H A L L E N G E • T E C H N O L O G Y For Exercises 71–74, let A, B, and C be the lengths of the three sides with X, Y, and Z as the corresponding angle measures. Write a program using the TI calculator to solve the given triangle. A B C X Z Y 71. B 5 45, C 5 57, and X 5 43° 72. B 5 24.5, C 5 31.6, and X 5 81.5° 73. A 5 29.8, B 5 37.6, and C 5 53.2 74. A 5 100, B 5 170, and C 5 250 SKILLS OBJECTIVES ■ ■ Find the area of a triangle in the SAS case. ■ ■ Find the area of a triangle in the SSS case. CONCEPTUAL OBJECTIVES ■ ■ Understand how to derive a formula for the area of a triangle (SAS case) using the Law of Sines. ■ ■ Understand how to derive a formula for the area of a triangle (SSS case) using the Law of Cosines. 7.3 THE AREA OF A TRIANGLE In Sections 7.1 and 7.2, we used the Law of Sines and the Law of Cosines to solve oblique triangles, which means to find all of the side lengths and angle measures.
eBook - PDFPrecalculus
Functions and Graphs
- Earl Swokowski, Jeffery Cole(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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. Given two sides and the included angle of a triangle, we can use the law of cosines to find the third side. We may then use the law of sines to find another angle of the triangle. Whenever this procedure is followed, it is best to find the angle opposite the shortest side, since that angle is always acute. In this way, we avoid the possibility of obtaining two solutions when solving a trigonometric equation involving that angle, as illustrated in the following example. EXAMPLE 1 Using the law of cosines (SAS) Solve n ABC , given a 5 5.0, c 5 8.0 , and b 5 77 8 . Solution The triangle is sketched in Figure 2. Since b is the angle between sides a and c , we begin by approximating b (the side opposite b ) as follows: b 2 5 a 2 1 c 2 2 2 ac cos b law of cosines 5 s 5.0 d 2 1 s 8.0 d 2 2 2 s 5.0 ds 8.0 d cos 77 8 substitute for a , c , and b 5 89 2 80 cos 77 8 < 71.0 simplify and approximate b < Ï 71.0 < 8.4 take the square root Let us find another angle of the triangle using the law of sines. In accor-dance with the remarks preceding this example, we will apply the law of sines and find a , since it is the angle opposite the shortest side a : sin a a 5 sin b b law of sines sin a 5 a sin b b solve for sin a < 5.0 sin 77 8 Ï 71.0 < 0.5782 substitute and approximate Since a is acute, a 5 sin 2 1 s 0.5782 d < 35.3 8 < 35 8 . Finally, since a 1 b 1 g 5 180 8 , we have g 5 180 8 2 a 2 b < 180 8 2 35 8 2 77 8 5 68 8 . ■ Given the three sides of a triangle, we can use the law of cosines to find any of the three angles.
eBook - PDFPrecalculus
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
The Law of Sines relates the sines of the angles to the sides as follows. Law of Sines Let ABC be a triangle with sides a, b, c . ( See Figure 1.) Then the following ratios hold: a _____ sin A = b _____ sin B = c _____ sin C The ratios can also be written as sin A _____ a = sin B _____ b = sin C _____ c Proof of Law of Sines Case 1 : A is acute. Referring to Figure 2, we have sin A = h __ b ⇒ h = b sin A sin B = h __ a ⇒ h = a sin B Equating these two expressions for h , we find that b sin A = a sin B Dividing by (sin A )(sin B ), we have b _____ sin B = a _____ sin A (1) Note that sin A and sin B are both always nonzero, because the angles of a triangle are greater than 0 ° and less than 180 ° . Next, referring to Figure 2, we have sin B = h ′ __ c ; sin C = h ′ __ b Solving each of these equations for hʹ and then equating the two expressions for hʹ gives c sin B = b sin C ⇒ c _____ sin C = b _____ sin B Combining this result with Equation (1) gives a _____ sin A = b _____ sin B = c _____ sin C Case 2 : A is obtuse. Referring to Figure 3, we have sin(180 ° − A ) = h __ b sin B = h __ a A B C b a c Figure 1 Note You can use the ratios in either form when solving a triangle, as long as you are consistent. b a c h h 9 C A B Figure 2 C A B a b c h h 9 Figure 3 The Law of Sines Objectives ■ Solve an oblique triangle using the Law of Sines. ■ Solve applied problems using the Law of Sines. 556 Chapter 7 Additional Topics in Trigonometry Using the difference identity for sine, sin(180 0 − A ) = sin A . Thus, sin A = h __ b ; sin B = h __ a Following the same procedure as in Case 1 gives: b _____ sin B = a _____ sin A Continue just as in Case 1 after Equation (1) to obtain a _____ sin A = b _____ sin B = c _____ sin C You can use the Law of Sines to solve an oblique triangle when you are given the following information about the angles and sides. ■ ASA: A side is common to the two angles (Figure 4). ■ AAS: A side is opposite to one of the angles (Figure 5).
eBook - PDF- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
6.1 Law of Sines 6.2 Law of Cosines 6.3 Areas of Triangles 6.4 Vectors 6.5 The Dot Product Chapter 6 Group Activity: The SSA Case and the Law of Cosines Chapter 6 Review 6 Additional Topics: Triangles and Vectors 349 ✩ Sections marked with a star may be omitted without loss of continuity. ✩ ✩ I I n Chapter 1 we introduced the trigonometric ratios to solve right triangles. We also saw that this could be very useful in solving applied problems, par- ticularly those involving measurement. But it’s unreasonable to assume that every situation that involves triangles will involve right triangles. So we will now return to solving triangles, but without being restricted to right triangles. We will establish the law of sines and law of cosines; they are the principal tools we’ll need to solve general triangles. We will then use these tools to solve a wide vari- ety of applied problems. We will also develop several formulas for calculating the area of a triangle. Velocity and force are examples of vector quantities. In Section 6.4, we intro- duce the concept of vector, and define vector addition and scalar multiplication. In Section 6.5 we discuss the dot product of vectors. We show how vectors and trigonometry can be used together to solve problems in physics and engineering. 6.1 LAW OF SINES • Deriving the Law of Sines • Solving ASA and AAS Cases • Solving the Ambiguous SSA Case Until now, we have considered only triangle problems that involved right trian- gles. We now turn to oblique triangles, that is, triangles that contain no right angle. Every oblique triangle is either acute (all angles are between 0° and 90°) or obtuse (one angle is between 90° and 180°; because the three angles always sum to 180°, there can’t be more than one greater than 90°). Figure 1 illustrates an acute trian- gle and an obtuse triangle.
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
663 10.1 Right-Triangle Applications 10.2 The Law of Sines and the Law of Cosines 10.3 Vectors in the Plane: A Geometric Approach 10.4 Vectors in the Plane: An Algebraic Approach 10.5 Parametric Equations 10.6 Introduction to Polar Coordinates 10.7 Curves in Polar Coordinates 10.8 DeMoivre’s Theorem The subject of trigonometry is an excellent example of a branch of mathematics . . . which was motivated by both practical and intellectual interests—surveying, map-making, and navigation on the one hand, and curiosity about the size of the universe on the other. With it the Alexandrian mathe-maticians triangulated the universe and rendered precise their knowledge about the Earth and the heavens. —Morris Klein in Mathematics in Western Culture (New York: Oxford University Press, 1953) This is our fifth and final chapter on trigonometry. Some of the subject matter here takes us back to the historical roots of the subject: the study of the relationships between the sides and angles of a triangle. In Section 10.1 we look at some applica-tions of right-triangle trigonometry that we didn’t do in Chapters 6 and 7. Section 10.2 presents the law of sines and the law of cosines. These relate the angles and lengths of the sides of arbitrary triangles. In Sections 10.3 and 10.4 we introduce the impor-tant topic of vectors, first from a geometric standpoint and then from an algebraic standpoint. In Sections 10.5 through 10.7 we expand upon some of the ideas in Chapters 1 and 3 on graphs, and equations, as we look at parametric equations and polar coordinates. Section 10.8 uses trigonometric identities and polar coordinates to present DeMoivre’s theorem, which we use to find the n distinct n th roots of a complex number. CHAPTER 10 Additional Topics in Trigonometry 10.1 RIGHT-TRIANGLE APPLICATIONS* We continue the work we began in Section 7.5 on right-triangle trigonometry.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method. 65. Writing In Exercise 64, the Law of Cosines was used to solve a triangle in the two-solution case of SSA. Can the Law of Cosines be used to solve the no-solution and single-solution cases of SSA? Explain. 66. HOW DO YOU SEE IT? To solve the triangle, would you begin by using the Law of Sines or the Law of Cosines? Explain. (a) A B a = 12 b = 16 c = 18 C (b) A B a = 18 b c C 35° 55° 67. Proof Use the Law of Cosines to prove each identity. (a) 1 2 bc(1 + cos A) = a + b + c 2 uni2219 -a + b + c 2 (b) 1 2 bc(1 - cos A) = a - b + c 2 uni2219 a + b - c 2 An engine has a seven-inch connecting rod fastened to a crank (see figure). 7 in. 1.5 in. x θ (a) Use the Law of Cosines to write an equation giving the relationship between x and θ . (b) Write x as a function of θ . (Select the sign that yields positive values of x.) (c) Use a graphing utility to graph the function in part (b). (d) Use the graph in part (c) to determine the total distance the piston moves in one cycle. 56. Mechanical Engineering Smart-foto/Shutterstock.com Copyright 2018 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. 3.3 Vectors in the Plane 278 Chapter 3 Additional Topics in Trigonometry Represent vectors as directed line segments. Write component forms of vectors. Perform basic vector operations and represent vector operations graphically. Write vectors as linear combinations of unit vectors.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
His works describe how to find areas of triangles, quadrilaterals, regular polygons with 3 to 12 sides, and circles, as well as surface areas and volumes of three-dimensional objects. HISTORICAL NOTE Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is Area = √s(s - a)(s - b)(s - c) where s = a + b + c 2 . Summarize (Section 8.2) 1. State the Law of Cosines (page 569). For examples of using the Law of Cosines to solve oblique triangles (SSS or SAS), see Examples 1 and 2. 2. Describe real-life applications of the Law of Cosines (page 571, Examples 3 and 4). 3. State Heron’s Area Formula (page 572). For an example of using Heron’s Area Formula to find the area of a triangle, see Example 5. 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. 8.2 Law of Cosines 573 8.2 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. GO DIGITAL Vocabulary and Concept Check In Exercises 1 and 2, fill in the blanks. 1. When solving an oblique triangle given three sides, use the ________ form of the Law of Cosines to solve for an angle. 2. When solving an oblique triangle given two sides and their included angle, use the ________ form of the Law of Cosines to solve for the remaining side. 3. State the alternative form of the Law of Cosines for b 2 = a 2 + c 2 - 2ac cos B. 4. State two other formulas, besides Heron’s Area Formula, that you can use to find the area of a triangle. Skills and Applications Using the Law of Cosines In Exercises 5–24, use the Law of Cosines to solve the triangle.
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