Sine and Cosine Rules

11 Key excerpts on "Sine and Cosine Rules"

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  Precalculus

  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 ◦ .

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  Maths: A Student's Survival Guide

  A Self-Help Workbook for Science and Engineering Students

  • Jenny Olive(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  We now have the Cosine Rule for any triangle. 150 Some trigonometry and geometry Figure 4.B.7 The Cosine Rule a 2 = b 2 + c 2 – 2 bc cos A (1) b 2 = c 2 + a 2 – 2 ac cos B (2) c 2 = a 2 + b 2 – 2 ab cos C (3) Notice also that if we put A = 90° in (1) above, we get Pythagoras’ Theorem for what is now a right-angled triangle. That is, we get a 2 = b 2 + c 2 because cos 90° = 0, so everything connects up as it should do. Here is an example of using the Cosine Rule to find a side of a triangle. We will use it to find a in ABC shown in Figure 4.B.8 . This triangle is another example of a case in which the Sine Rule will not give us what we want. This is because the known facts slot into it in such a way that every possible equation has two unknowns. We would have a sin 72° = 10 sin B = 8 sin C which is no use. Using the Cosine Rule, we have a 2 = b 2 + c 2 – 2 bc cos A . Substituting the known values, this gives us a 2 = 64 + 100 – 160 cos 72° so a = 10.7 to 1 d.p. If we want to find the angles of a triangle using the Cosine Rule, it will pay us to rearrange the three formulas. For example, we have a 2 = b 2 + c 2 – 2 bc cos A so 2 bc cos A = b 2 + c 2 – a 2 . Rearranging this gives us cos A = b 2 + c 2 – a 2 2 bc , (1) cos B = c 2 + a 2 – b 2 2 ca , (2) cos C = a 2 + b 2 – c 2 2 ab , (3) shifting the letters round again in turn to give the other two formulas. 4.B Widening the field in trigonometry 151 Figure 4.B.8 We take the triangle from the beginning of this section to show the use of the Cosine Rule to find its angles. It has sides of 5 cm, 7 cm and 9 cm and I show it again in Figure 4.B.9. We will now find the angles A , B and C . I want the angles to go in this way, which is why my lettering of the triangle isn’t the usual one. Using the Cosine Rule to find A , we have cos A = b 2 + c 2 – a 2 2 bc = 49 + 81 – 25 126 so cos A = 105 126 and A = 33.6° to 1 d.p.

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  Algebra and Trigonometry

  • 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 .

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  Introduction to Trigonometry, An

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 5 Common Laws & Formulas in Trigonometry 1. Law of sines A triangle In trigonometry, the law of sines (also known as the sine law , sine formula , or sine rule ) is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law, where a , b , and c are the lengths of the sides of a triangle, and A , B , and C are the opposite angles (see the figure above). Sometimes the law is stated using the reciprocal of this equation: The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case . ________________________ WORLD TECHNOLOGIES ________________________ The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines. Examples The following are examples of how to solve a problem using the law of sines: Given: side a = 20, side c = 24, and angle C = 40° Using the law of sines, we conclude that Or another example of how to solve a problem using the law of sines: If two sides of the triangle are equal to R and the length of the third side, the chord, is given as 100 feet and the angle C opposite the chord is given in degrees, then and Numeric problems Like the law of cosines, although the law of sines is mathematically true, it has problems for numeric use. Much precision may be lost if an arcsine is computed when the sine of an angle is close to one. ________________________ WORLD TECHNOLOGIES ________________________ Some applications • The sine law can be used to prove the angle sum identity for sine when α and β are each between 0 and 90 degrees.

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  Precalculus

  Building Concepts and Connections 2E

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

    (Publisher)

  H igh precision instruments are used in sports to make accurate measurements in competitions. For instance, a small computer in the measuring device uses trigonometric calculations to compute the distance an athlete throws a discus. See Exercise 37 in Section 7.2. In this chapter, we will explore additional methods for solving triangles, investigate vectors, and look at alternate ways to represent complex numbers using a trigonometric form and to represent points in the plane using polar form. Chapter 7 Outline p 0 1 2 3 4 2 3 p 6 7 p 3 5 p 6 5 p 3 2 p 6 11 p 3 4 p 2 p 3 p 3, 3 p ( ( 6 p 7.1 The Law of Sines 7.2 The Law of Cosines 7.3 Polar Coordinates 7.4 Graphs of Polar Equations 7.5 Vectors 7.6 Dot Product of Vectors 7.7 Trigonometric Form of a Complex Number Additional Topics in Trigonometry 7 . 1 7.1 The Law of Sines 555 In Chapter 5 you learned how to use right triangles to solve a variety of problems. Many interesting problems involve oblique triangles , which are triangles not containing a right angle. In this section and the next, you will learn techniques for solving problems using oblique triangles. Throughout these two sections, we will use the standard notation for triangles. The angles are labeled A , B , and C , and the sides opposite those angles are labeled a , b , and c , respectively, as shown in Figure 1. 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.

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  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(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. In the next section we will discuss the law of cosines and show how it can be used to find the remaining parts of an oblique triangle when given the following: (1) two sides and the angle between them (SAS) (2) three sides (SSS) The law of sines cannot be applied directly to the last two cases. The law of sines can also be written in the form a sin a 5 b sin b 5 c sin g . Instead of memorizing the three formulas associated with the law of sines, it may be more convenient to remember the following statement, which takes all of them into account. In examples and exercises involving triangles, we shall assume that known lengths of sides and angles have been obtained by measurement and hence are approximations to exact values. Unless directed otherwise, when finding parts of triangles we will round off answers according to the following rule: If known sides or angles are stated to a certain accuracy, then unknown sides or angles should be calculated to the same accuracy. To illustrate, if known sides are stated to the nearest 0.1, then unknown sides should be cal-culated to the nearest 0.1. If known angles are stated to the nearest 10 9 , then unknown angles should be calculated to the nearest 10 9 . Similar remarks hold for accuracy to the nearest 0.01, 0.1°, and so on. EXAMPLE 1 Using the law of sines (ASA) Solve n ABC , given a 5 48 8 , g 5 57 8 , and b 5 47 . Solution The triangle is sketched in Figure 2. Since the sum of the angles of a triangle is 180°, b 5 180 8 2 57 8 2 48 8 5 75 8 .

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  Trigonometry

  • Cynthia Y. Young(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  In this case, start by finding the measure of the largest angle (opposite the longest side) using the Law of Cosines. Then use the Law of Sines to find either of the remaining two angle measures. Finally, find the third angle measure using the fact that the three angles in a triangle always sum to 1808. EXAMPLE 3 Using the Law of Cosines to Solve a Triangle (SSS) Solve the triangle a 5 8, b 5 6, and c 5 7. Solution: STEP 1 Identify the largest angle, which is a. Use the Law of Cosines that involves a. a 2 5 b 2 1 c 2 2 2 bc cos a Let a 5 8, b 5 6, and c 5 7. 8 2 5 6 2 1 7 2 2 2 1 6 21 7 2 cos a Simplify and isolate cos a. cos a 5 6 2 1 7 2 2 8 2 2 1 6 21 7 2 5 0.25 Apply the inverse cosine function. a 5 cos 21 1 0.25 2 Approximate with a calculator. a < 76° Round to the nearest degree. STEP 2 Find either of the remaining angle measures. To solve for acute angle b: Use the Law of Sines. sin a a 5 sin b b Isolate sin b. sin b 5 b sin a a Let a 5 8, b 5 6, and a 5 76°. sin b 5 6 sin 76° 8 Use the inverse sine function. b 5 sin 21 a 6 sin 76° 8 b Approximate with a calculator. b < 47° STEP 3 Find the measure of the third angle g. The sum of the angle measures is 180°. 76° 1 47° 1 g 5 180° Solve for g. g < 57° YOUR TURN Solve the triangle a 5 5, b 5 7, and c 5 8. ▼ In the next example, instead of immediately substituting values into the Law of Cosines equation, we will solve for the angle measure in general, and then substitute in values. ▼ A N S W E R a < 38°, b < 60°, and g < 82° 7.2 The Law of Cosines 359 360 CHAPTER 7 Applications of Trigonometry: Triangles and Vectors EXAMPLE 4 Using the Law of Cosines in an Application (SSS) In recent decades, many people have come to believe that an imaginary area called the Bermuda Triangle, located off the southeastern Atlantic coast of the United States, has been the site of a high incidence of losses of ships, small boats, and aircraft over the centuries.

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  Precalculus, Enhanced Edition

  • David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Find (There are two answers.) Hint: Solve the given equation for c 2 . 56. In this section we have seen that the cosines of the angles in a triangle can be expressed in terms of the lengths of the sides. For instance, for cos A in ^ ABC , we obtained cos A ( b 2 c 2 a 2 ) 2 bc. This exercise shows how to derive corresponding expressions for the sines of the angles. For ease of notation in this exercise, let us agree to use the letter T to denote the following quantity: T 2( a 2 b 2 b 2 c 2 c 2 a 2 ) ( a 4 b 4 c 4 ) Then the sines of the angles in ^ ABC are given by In the steps that follow, we’ll derive the first of these three formulas, the derivations for the other two being entirely similar. (a) In ^ ABC , why is the positive root always appropriate in the formula sin A (b) In the formula in part (a), replace cos A by ( b 2 c 2 a 2 ) 2 bc and show that the result can be written (c) On the right-hand side of the equation in part (b), carry out the indicated multiplication. After combining like terms, you should obtain sin A 2 bc , as required. 57. In the two easy steps that follow, we derive the law of sines by using the formulas obtained in Exercise 56. (Since the formulas in Exercise 56 were obtained using the law of cosines, we are, in essence, showing how to derive the law of sines from the law of cosines.) (a) Use the formulas in Exercise 56 to check that each of the three fractions (sin A ) a , (sin B ) b , and (sin C ) c is equal to 2 abc. (b) Conclude from part (a) that (sin A ) a (sin B ) b (sin C ) c 58. In this exercise you are going to use the law of cosines and the law of sines to determine the area of the shaded equilat-eral triangle in Figure A. Begin by labeling points, as shown in Figure B. 1 T 1 T sin A 2 4 b 2 c 2 ( b 2 c 2 a 2 ) 2 2 bc 2 1 cos 2 A ? sin A 1 T 2 bc sin B 1 T 2 ac sin C 1 T 2 ab C . cos A a cos B b cos C c a 2 b 2 c 2 2 abc 10.2 The Law of Sines and the Law of Cosines 695 Copyright 201 Cengage Learning.

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  Algebra and Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  We choose the smaller angle first to avoid the ambiguity with the Law of Sines. Alternatively, if we wanted to solve for the obtuse angle first, we could have used the Law of Cosines to solve for α. STUDY TIP Although the Law of Sines can sometimes lead to the ambiguous case, the Law of Cosines never leads to the ambiguous case. EXAMPLE 2 Using the Law of Cosines in an Application (SAS) In an AKC (American Kennel Club)-sanctioned field trial, a judge sets up a mark (bird) that requires the dog to swim across a body of water (the dogs are judged on how closely they adhere to the straight line to the bird, not the time it takes to retrieve the bird). The judge is trying to calculate how far the dog would have to swim to this mark, so she walks off the two legs across the land and measures the angle as shown in the figure. How far will the dog swim from the starting line to the bird? Solution Label the triangle. Use the Law of Cosines. c 2 = a 2 + b 2 − 2ab cos γ Let a = 176, b = 152, and γ = 117°. c 2 = 176 2 + 152 2 − 2(176)(152) cos 117° Use a calculator to approximate the right side. c 2 ≈ 78370.3077 Solve for c and round to the nearest yard (three significant digits). c ≈ 280 yd 152 yd Pond 176 yd 117º b = 152 yd Pond a = 176 yd  = 117º c 8.2 The Law of Cosines 793 Case 4: Solving Oblique Triangles (SSS) We now solve oblique triangles when all three side lengths are given (the SSS case). In this case, start by finding the measure of the largest angle (opposite the longest side) using the Law of Cosines. Then use the Law of Sines to find either of the remaining two angle measures. Finally, find the third angle measure using the fact that the three angles in a triangle always sum to 180°. Video EXAMPLE 3 Using the Law of Cosines to Solve a Triangle (SSS) Solve the triangle a = 8, b = 6, and c = 7. Solution STEP 1 Identify the largest angle, which is α. Use the Law of Cosines that involves α. a 2 = b 2 + c 2 − 2bc cos α Let a = 8, b = 6, and c = 7.

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  Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  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 7.3 The Area of a Triangle 367 7.3 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 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. Now, we use these laws to derive formulas for the area of a triangle (SAS and SSS cases). Our starting point for both cases is the standard formula for the area of a triangle: A = 1 __ 2 bh 7.3.1 The Area of a Triangle (SAS Case) 7.3.1 Skill Find the area of a triangle in the SAS case. 7.3.1 Conceptual Understand how to derive a formula for the area of a triangle (SAS case) using the Law of Sines. We now can use the general formula for the area of a triangle and the Law of Sines to develop a formula for the area of a triangle when the length of two sides and the measure of the angle between them are given.

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  Precalculus with Limits

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

    (Publisher)

  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° 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. 416 Chapter 6 Additional Topics in Trigonometry 6.3 Vectors in the Plane GO DIGITAL 6.3 Vectors in the Plane 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. Find direction angles of vectors. Use vectors to model and solve real-life problems. Introduction Quantities such as force and velocity involve both magnitude and direction and cannot be completely characterized by a single real number. To represent such a quantity, you can use a directed line segment, as shown in Figure 6.10. The directed line segment PQ \ has initial point P and terminal point Q. Its magnitude (or length) is denoted by  PQ \  and can be found using the Distance Formula. Initial point P Q PQ Terminal point Figure 6.10 Figure 6.11 Two directed line segments that have the same magnitude and direction are equivalent. For example, the directed line segments in Figure 6.11 are all equivalent. The set of all directed line segments that are equivalent to the directed line segment PQ \ is a vector v in the plane, written v = PQ \ . Vectors are denoted by lowercase, boldface letters such as u, v, and w. EXAMPLE 1 Showing That Two Vectors Are Equivalent Show that u and v in Figure 6.12 are equivalent.

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