Mathematics
Law of Cosines
The Law of Cosines is a mathematical formula used to find the length of a side of a triangle when the lengths of the other two sides and the angle between them are known. It is an extension of the Pythagorean theorem and is particularly useful for solving non-right-angled triangles. The formula is expressed as c^2 = a^2 + b^2 - 2ab*cos(C), where c is the unknown side, a and b are the known sides, and C is the angle between sides a and b.
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9 Key excerpts on "Law of Cosines"
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)
- 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.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
We then can use either the Law of Sines or the Law of Cosines to find the measure of a second angle. 7.2 The Law of Cosines 359 EXAMPLE 1 Using the Law of Cosines to Solve a Triangle (SAS) Solve the triangle a = 13, c = 6.0, and β = 20°. Solution Two side lengths and the measure of the angle between them are given (SAS). Notice that the Law of Sines cannot be used because it requires knowledge of at least one angle measure and the length of the side opposite that angle. STEP 1 Find b. Use the Law of Cosines that involves β. b 2 = a 2 + c 2 − 2ac cos β Let a = 13, c = 6.0, and β = 20°. b 2 = 13 2 + 6 2 − 2(13)(6) cos 20° Evaluate the right side using a calculator. b 2 ≈ 58.40795 Solve for b. b ≈ ±7.6425 Round to two significant digits; The length of a side can be only positive. b ≈ 7.6 STEP 2 Find the acute angle γ. Use the Law of Sines to find the smaller angle γ. sin γ _____ c = sin β _____ b Isolate sin γ. sin γ = c sin β ______ b Let b = 7.6, c = 6.0, and β = 20°. sin γ ≈ 6 sin 20° ________ 7.6 Use the inverse sine function. γ ≈ sin −1 ( 6 sin 20° ________ 7.6 ) Evaluate the right side with a calculator. γ ≈ 15.66521° Round to the nearest degree. γ ≈ 16° STEP 3 Find α. The three angle measures must sum to 180°. α + 20° + 16° ≈ 180° Solve for α. α ≈ 144° Your Turn Solve the triangle b = 4.2, c = 1.8, and α = 35°. Answer a ≈ 2.9, γ ≈ 21°, and β ≈ 124° c = 6.0 a = 13 = 20º b 360 CHAPTER 7 Applications of Trigonometry: Triangles and Vectors Notice the steps we took in solving a SAS triangle: 1. Find the length of the side opposite the given angle using the Law of Cosines. 2. Solve for the smaller angle (which has to be acute) using the Law of Sines. 3. Solve for the larger angle using properties of triangles. You may be thinking, would it matter if we solved for α before solving for γ ? Yes, it does matter—in this problem, you cannot solve for α by the Law of Sines before finding γ.
eBook - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
b 2 − x 2 = a 2 − (c − x) 2 Multiply out the squared binomial on the right. b 2 − x 2 = a 2 − ( c 2 − 2cx + x 2 ) Eliminate the parentheses. b 2 − x 2 = a 2 − c 2 + 2cx − x 2 Add x 2 to both sides. b 2 = a 2 − c 2 + 2cx Isolate a 2 . a 2 = b 2 + c 2 − 2cx Notice that cos α = x __ b . Let x = b cos α. a 2 = b 2 + c 2 − 2bc cos α α β γ a b c α β a b h c 1 2 c – x x γ 420 CHAPTER 4 Trigonometric Functions of Angles The Pythagorean theorem can thus be regarded as a special case of the Law of Cosines. Note: If we instead drop the perpendicular line segment with length h from the angle α or the angle β to the opposite side, we can derive the other two forms of the Law of Cosines. b 2 = a 2 + c 2 − 2ac cos β and c 2 = a 2 + b 2 − 2ab cos γ The Law of Cosines For a triangle with sides a, b, and c, and opposite angles α, β, and γ, the following is true: a 2 = b 2 + c 2 − 2bc cos α b 2 = a 2 + c 2 − 2ac cos β c 2 = a 2 + b 2 − 2ab cos γ It is important to note that the Law of Cosines can be used to find unknown side lengths or angle measures. As long as three of the four variables in any of the equations are known, the fourth can be calculated. Notice that in the special case of a right triangle (say, α = 90°), a 2 = b 2 + c 2 − 2bc cos 90° ⏟ 0 one of the forms of the Law of Cosines reduces to the Pythagorean theorem: a 2 ⏟ hyp = b 2 ⏟ leg + c 2 ⏟ leg c b γ α β a STUDY TIP The Pythagorean theorem is a special case of the Law of Cosines. Concept Check TRUE OR FALSE The Pythagorean theorem only applies to right triangles, whereas the Law of Sines and the Law of Cosines can be applied to acute and obtuse triangles. Answer: True Case 3: Solving Oblique Triangles (SAS) We can now solve SAS triangle problems, where the angle between two known sides is given. We start by using the Law of Cosines to solve for the length of the side opposite the given angle. We then can use either the Law of Sines or the Law of Cosines to find the measure of a second angle.
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)
- 2021(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. 8.2 Law of Cosines 575 53. Surveying A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of the largest angle. 54. Distance Two ships leave a port at 9 a.m. One travels at a bearing of N 53° W at 12 miles per hour, and the other travels at a bearing of S 67° W at s miles per hour. (a) Use the Law of Cosines to write an equation that relates s and the distance d between the two ships at noon. (b) Find the speed s that the second ship must travel so that the ships are 43 miles apart at noon. 55. Geometry A triangular parcel of land has sides of lengths 200 feet, 500 feet, and 600 feet. Use Heron’s Area Formula to find the area of the parcel. 56. Geometry A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70°. Use the Law of Cosines and Heron’s Area Formula to find the area of the parking lot. 70° 70 m 100 m 57. Geometry You want to buy a triangular lot measuring 510 yards by 840 yards by 1120 yards. The price of the land is $2000 per acre. How much does the land cost? (Hint: 1 acre = 4840 square yards) 58. Geometry You want to buy a triangular lot measuring 1350 feet by 1860 feet by 2490 feet. The cost of the land is $62,000. What is the price of the land per acre? (Hint: 1 acre = 43,560 square feet) Exploring the Concepts True or False? In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. 59. In Heron’s Area Formula, s is the average of the lengths of the three sides of the triangle.
eBook - PDF- Ron Larson(Author)
- 2021(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. 6.2 Law of Cosines 415 53. Surveying A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of the largest angle. 54. Distance Two ships leave a port at 9 a.m. One travels at a bearing of N 53° W at 12 miles per hour, and the other travels at a bearing of S 67° W at s miles per hour. (a) Use the Law of Cosines to write an equation that relates s and the distance d between the two ships at noon. (b) Find the speed s that the second ship must travel so that the ships are 43 miles apart at noon. 55. Geometry A triangular parcel of land has sides of lengths 200 feet, 500 feet, and 600 feet. Use Heron’s Area Formula to find the area of the parcel. 56. Geometry A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70°. Use the Law of Cosines and Heron’s Area Formula to find the area of the parking lot. 70° 70 m 100 m 57. Geometry You want to buy a triangular lot measuring 510 yards by 840 yards by 1120 yards. The price of the land is $2000 per acre. How much does the land cost? (Hint: 1 acre = 4840 square yards) 58. Geometry You want to buy a triangular lot measuring 1350 feet by 1860 feet by 2490 feet. The cost of the land is $62,000. What is the price of the land per acre? (Hint: 1 acre = 43,560 square feet) Exploring the Concepts True or False? In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. 59. In Heron’s Area Formula, s is the average of the lengths of the three sides of the triangle.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
In a similar manner, construct an altitude h from vertex B to side AC (extended in the obtuse triangle), as shown at the left. Then you have sin A = h c h = c sin A and sin C = h a h = a sin C. Equating these two values of h, you have a sin C = c sin A or a sin A = c sin C . By the Transitive Property of Equality, a sin A = b sin B = c sin C . Law of Cosines (p. 271) Standard Form Alternative Form a 2 = b 2 + c 2 - 2bc cos A cos A = b 2 + c 2 - a 2 2bc b 2 = a 2 + c 2 - 2ac cos B cos B = a 2 + c 2 - b 2 2ac c 2 = a 2 + b 2 - 2ab cos C cos C = a 2 + b 2 - c 2 2ab LAW OF TANGENTS Besides the Law of Sines and the Law of Cosines, there is also a Law of Tangents, developed by French mathematician François Viète (1540–1603). The Law of Tangents follows from the Law of Sines and the sum-to-product formulas for sine and is defined as a + b a - b = tan[(A + B)H208622] tan[(A - B)H208622] . The Law of Tangents can be used to solve a triangle when two sides and the included angle (SAS) are given. Before the invention of calculators, it was easier to use the Law of Tangents to solve the SAS case instead of the Law of Cosines because the computations by hand were not as tedious. a c A b C A is acute. h B a c B A b C A is obtuse. h 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. 309 Proof To prove the first formula, consider the triangle at the left, which has three acute angles. Note that vertex B has coordinates (c, 0). Furthermore, C has coordinates (x, y), where x = b cos A and y = b sin A.
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