Triangle trigonometry

11 Key excerpts on "Triangle trigonometry"

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  Introductory Technical Mathematics

  • John Peterson, Robert Smith(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  F u n d a m e n t a l s o f Tr i g o n o m e t r y S E C T I O N V I I Copyright 2019 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. 838 OBJECTIVES After studying this unit you should be able to ■ identify the sides of a right triangle with reference to any angle. ■ state the ratios of the six trigonometric functions in relation to given triangles. ■ find functions of angles given in decimal degrees and degrees, minutes, and seconds. ■ find angles in decimal degrees and degrees, minutes, and seconds of given functions. Trigonometry is the branch of mathematics that is used to compute unknown angles and sides of triangles. The word trigonometry is derived from the Greek words for triangle and measurement. Trigonometry is based on the principles of geometry. Many problems require the use of geometry and trigonometry. As with geometry, much in our lives depends on trigonometry. The methods of trigonometry are used in constructing buildings, roads, and bridges. Trigonometry is used in the design of automobiles, trains, airplanes, and ships. The machines that produce the manufactured products we need could not be made without the use of trigonometry. A knowledge of trigonometry and the ability to apply the knowledge in actual occupational uses is required in many skilled trades. Machinists, surveyors, drafters, electricians, and electronics technicians are a few of the many occupations in which trigonometry is a requirement. Practical problems are often solved by using a combination of elements of algebra, geometry, and trigonometry.

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  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 5 Trigonometry and Trigonometric Functions 5.1 Introduction

  The word trigonometry is derived from two Greek words, together meaning measuring the sides of a triangle . The subject was originally developed to solve geometric problems involving triangles. One of its uses lies in determining heights and distances , which are not easy to measure otherwise. It has been very useful in surveying , navigation , and astronomy . Applications have now further widened.

  At school level, in geometry, we have studied the definitions of trigonometric ratios of acute angles in terms of the ratios of sides of a right-angled triangle.

  Note that in the right-angled triangle OAR , if the lengths of the sides are respectively denoted by B (for base), P (for perpendicular), and H (for hypotenuse), as shown in Figure 5.1 , then the angle θ (in degrees) is an acute angle (i.e., 0° < θ < 90°). It is for such angle(s) that we have defined trigonometric ratios in earlier classes.1

  Figure 5.1 Right angled triangle defining trigonometric ratios.

  Now, in our study of trigonometry, it is required to extend the notion of an angle in such a way that its measure can be of any magnitude and sign . Once this is done, the trigonometric ratios are defined for angles of all magnitudes and sign. Finally, by identifying these magnitudes and signs of angles, with real numbers, we say that the trigonometric ratios of directed angles represent trigonometric functions of real variables

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  Barron's Math 360: A Complete Study Guide to Pre-Calculus with Online Practice

  • Lawrence S. Leff, Christina Pawlowski-Polanish, Barron's Educational Series, Elizabeth Waite(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  STUDY UNIT III

  TRIGONOMETRIC ANALYSIS

  Passage contains an image

  9 TRIGONOMETRY

  WHAT YOU WILL LEARN

  Trigonometry means “measurement of triangles.” The study of trigonometry arose from the ancient need to understand the relationships between the sides and angles of triangles. With the development of calculus, trigonometry progressed from the study of ratios within right triangles to trigonometric functions that could be used to better represent the circular and repeating patterns of behavior that characterize a wide range of physical phenomena in the real world.

  This chapter progresses from considering acute angles in right triangles to a more general view of angles as rotations about the origin in the coordinate plane. By fixing the vertex of such an angle at the origin and keeping one side of the angle aligned with the positive x-axis, we can give meaning to trigonometric functions of angles greater than 90° and less than 0°.

  LESSONS IN CHAPTER 9

  •Lesson 9-1: Degree and Radian Measures

  •Lesson 9-2: Right-Triangle trigonometry

  •Lesson 9-3: The General Angle

  •Lesson 9-4: Working with Trigonometric Functions

  •Lesson 9-5: Trigonometric Functions of Special Angles

  Lesson 9-1: Degree and Radian Measures

  KEY IDEAS

  Angle measures can be expressed in units of degrees or in real-number units called radians. Degrees are based on fractional parts of a circular revolution. Radian measure compares the length of an arc that a central angle of a circle cuts off to the radius of the circle. The Greek letter θ (theta) is commonly used to represent an angle of unknown measure.

  MEASURING ANGLES IN DEGREES AND MINUTES

  One degree, denoted as 1°, is of one complete revolution about a fixed point.

  Each of the 60 equal parts of a degree is called a minute

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  Algebra & Trig

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

    (Publisher)

  Use fundamental trigonometric identities. Use trigonometric functions to model and solve real-life problems. The Six Trigonometric Functions This section introduces the trigonometric functions from a right triangle perspective. Consider the right triangle shown below, in which one acute angle is labeled θ . Relative to the angle θ , the three sides of the triangle are the hypotenuse, the opposite side (the side opposite the angle θ ), and the adjacent side (the side adjacent to the angle θ ). Hypotenuse θ Side adjacent to θ Side opposite θ Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle θ . sine cosecant cosine secant tangent cotangent Abbreviations for these six functions are sin, csc, cos, sec, tan, and cot, respectively. In the definitions below, 0° < θ < 90° (θ lies in the first quadrant). For such angles, the value of each trigonometric function is positive. Right Triangle trigonometry has many real-life applications. For example, in Exercise 68 on page 444, you will use right Triangle trigonometry to analyze the height of a helium-filled balloon. Right Triangle Definitions of Trigonometric Functions Let θ be an acute angle of a right triangle. The six trigonometric functions of the angle θ are defined below. (Note that the functions in the second column are the reciprocals of the corresponding functions in the first column.) sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. opp = the length of the side opposite θ adj = the length of the side adjacent to θ hyp = the length of the hypotenuse Georg Joachim Rheticus (1514–1576) was the leading Teutonic mathematical astronomer of the sixteenth century. He was the first to define the trigonometric functions as ratios of the sides of a right triangle.

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  4.3.1 Conceptual Understand that right triangle definitions of trigonometric functions for acute angles are consistent with definitions of trigonometric functions for all angles in the Cartesian plane. To define the trigonometric functions in the Cartesian plane, let us start with an acute angle θ in standard position. Choose any point (x, y) on the terminal side of the angle as long as it is 388 CHAPTER 4 Trigonometric Functions of Angles not the vertex (the origin). A right triangle can be drawn so that the right angle is made when a perpendicular segment connects the point (x, y) to the x-axis. Notice that the side opposite θ has length y and the other leg of the right triangle has length x. Words Math The distance r from the origin (0, 0) to the point (x, y) can be found using r = √ _______________ (x − 0) 2 + ( y − 0) 2 r = √ _ x 2 + y 2 the distance formula. Since r is a distance, it is always positive. r > 0 Using our first definition of trigonometric functions in terms of right triangle ratios (Section 4.2), we say that sin θ = opposite _________ hypotenuse . From this picture we see that the sine function can also be defined by the relation sin θ = y _ r . Similar reasoning holds for all six trigonometric functions and leads us to the second definition of the trigonometric functions, in terms of ratios of coordinates and distances in the Cartesian plane. (x, y) θ x y (x, y) θ x y r x y (x, y) θ x y r x y Let (x, y) be any point, other than the origin, on the terminal side of an angle θ in standard position. Let r be the distance from the point (x, y) to the origin; then the six trigonometric functions are defined as sin θ = y _ r cos θ = x _ r tan θ = y _ x (x ≠ 0) csc θ = r _ y ( y ≠ 0) sec θ = r _ x (x ≠ 0) cot θ = x _ y ( y ≠ 0) where r = √ _ x 2 + y 2 , or x 2 + y 2 = r 2 .

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  Pre-Calculus All-in-One For Dummies

  Book + Chapter Quizzes Online

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  In this section, you see three very important ratios in right triangles — sine, cosine, and tangent — as well as three not-so-vital but still important ratios — cosecant, secant, and cotangent. These ratios are all functions, where an angle is the input, and a real number is the output. Each function looks at an angle of a right triangle, known or unknown, and then uses the definition of its specific ratio to help you find missing information in the triangle quickly and easily. To round out this section, I show you how to use inverse trig functions to solve for unknown angles in a right triangle. Following the sine In a right triangle, the sine of an acute angle named theta is defined as the ratio of the length of the opposite leg to the length of the hypotenuse. In symbols, you write sin . Here’s what the ratio looks like: sin opposite hypotenuse . FIGURE 8-1: Angles co-terminal with a 360 angle. 176 UNIT 3 The Essentials of Trigonometry In order to find the sine of an angle, you must know the lengths of the opposite side and the hypotenuse. You will be given the lengths of two sides, but if the two sides aren’t the ones you need to find a certain ratio, you can use the Pythagorean Theorem to find the missing one. Q. Find the sine of angle F (sin F) in Figure 8-2. A. Follow these steps: 1. Identify the hypotenuse. Where’s the right angle? It’s R, so side r, across from it, is the hypotenuse. You can label it “Hyp.” 2. Locate the opposite side. Look at the angle in question, which is F here. Which side is across from it? Side f is the opposite leg. You can label it “Opp.” 3. Label the adjacent side. The only side that’s left, side k, has to be the adjacent leg. You can label it “Adj.” 4. Locate the two sides that you use in the trig ratio. Because you are finding the sine of F , you need the opposite side and the hypotenuse. For this triangle, leg leg hypotenuse 2 2 2 becomes f k r 2 2 2 .

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  Algebra and Trigonometry

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

    (Publisher)

  6.3 Solving Right Triangles 535 6.3.3 Solving a Right Triangle Given the Lengths of Two Sides 6.3.3 Skill Solve right triangles given two side lengths. 6.3.3 Conceptual Understand that the trigonometric inverse keys on a calculator can be used to approximate the measure of an angle, given its trigonometric function value. When solving a right triangle, we already know that one angle has measure 90°. Let us now consider the case when the lengths of two sides are given. In this case, the third side can be found using the Pythagorean theorem. If we can determine the measure of one of the acute angles, then we can find the measure of the third acute angle using the fact that the sum of the three angle measures in a triangle is 180°. How do we find the measure of one of the acute angles? Since we know the side lengths, we can use right triangle ratios to determine the trigonometric function (sine, cosine, or tangent) values and then ask ourselves: What angle corresponds to that value? Sometimes, we may know the answer exactly. For example, if we determine that sin θ = 1 _ 2 , then we know that the acute angle θ is 30° because sin 30° = 1 _ 2 . Other times we may not know the corresponding angle, such as sin θ = 0.9511. Calculators have three keys ( sin −1 , cos −1 , and tan −1 ) that help us determine the unknown angle. For example, a calculator can be used to assist us in finding what angle θ corresponds to sin θ = 0.9511. sin −1 (0.9511) = 72.00806419 At first glance, these three keys might appear to yield the reciprocal; however, the −1 superscript corresponds to an inverse function. We will learn more about inverse trigonometric functions in Chapter 7, but for now we will use these three calculator keys to help us solve right triangles. EXAMPLE 3 Using a Calculator to Determine an Acute Angle Measure Use a calculator to find θ. Round answers to the nearest degree. a. cos θ = 0.8734 b.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 520 CHAPTER 11 ■ INTRODUCTION TO TRIGONOMETRY Unless otherwise noted, all content on this page is © Cengage Learning. 10. For (not shown), suppose you know that , , and . a) Explain why you do not need to apply the Law of Sines or the Law of Cosines to find the measure of . b) Find . In Exercises 11 to 14, find the area of each triangle shown. Give the answer to the nearest tenth of a square unit. 11. 12. 13. 14. In Exercises 15 and 16, find the area of the given figure. Give the answer to the nearest tenth of a square unit. 15. 16. In Exercises 17 to 22, use a form of the Law of Sines to find the measure of the indicated side or angle. Angle measures should be found to the nearest degree and lengths of sides to the nearest tenth of a unit. 17. 18. 19. 20. 21. 22. In Exercises 23 to 28, use a form of the Law of Cosines to find the measure of the indicated side or angle. Angle measures should be found to the nearest degree and lengths of sides to the nearest tenth of a unit. 23. 24. 25. 26. 27. 28. In Exercises 29 to 34, use the Law of Sines or the Law of Cosines to solve each problem. Angle measures should be found to the nearest degree and areas and distances to the nearest tenth of a unit. 29. A triangular lot has street dimensions of 150 ft and 180 ft and an included angle of 80° for these two sides. a) Find the length of the remaining side of the lot. b) Find the area of the lot in square feet. 30. Phil and Matt observe a balloon. They are 500 ft apart, and their angles of observation are 47° and 65°, as shown. Find the distance x from Matt to the balloon. 31. A surveillance aircraft at point C sights an ammunition warehouse at A and enemy headquarters at B through the angles indicated. If points A and B are 10,000 m apart, what is the distance from the aircraft to enemy headquarters? 32. Above one room of a house the rafters meet as shown.

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  Trigonometry

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

    (Publisher)

  The trigonometric function values for the three special angle measures (30°, 45°, and 60°) are summarized in the following table: Trigonometric Function Values for Special Angles (30°, 45°, and 60°)  sin  cos  tan  cot  sec  csc  30° 1 _ 2 √ _ 3 _ 2 √ _ 3 _ 3 √ _ 3 2 √ _ 3 _ 3 2 45° √ _ 2 _ 2 √ _ 2 _ 2 1 1 √ _ 2 √ _ 2 60° √ _ 3 _ 2 1 _ 2 √ _ 3 √ _ 3 _ 3 2 2 √ _ 3 _ 3 It is important to learn the special values in red for the sine and cosine functions. All other values in the table can be found through reciprocals or quotients of these two functions. Remember that the tangent function is the ratio of the sine to cosine functions. sin θ = opposite _ hypotenuse cos θ = adjacent _ hypotenuse tan θ = sin θ _ cos θ = ( opposite _ hypotenuse ) _____________ ( adjacent _ hypotenuse ) = opposite _ adjacent STUDY TIP If you memorize the values for sine and cosine for the angles given in the table, then the other trigonometric function values in the table can be found using the quotient and reciprocal identities. STUDY TIP SOHCAHTOA: • Sine is Opposite over Hypotenuse • Cosine is Adjacent over Hypotenuse • Tangent is Opposite over Adjacent 1.4 Evaluating Trigonometric Functions: Exactly and with Calculators 39 1.4.2 Using Calculators to Evaluate (Approximate) Trigonometric Function Values 1.4.2 Skill Evaluate (approximate) trigonometric functions using a calculator. 1.4.2 Conceptual Understand the difference between evaluating trigonometric functions exactly and using a calculator. We will now turn our attention to using calculators to evaluate trigonometric functions, which sometimes results in an approximation. Scientific and graphing calculators have buttons for the sine (sin), cosine (cos), and tangent (tan) functions. Let us start with what we already know and confirm it with our calculators. STUDY TIP Make sure your calculator is set in degrees (DEG) mode. EXAMPLE 3 Evaluating Trigonometric Functions with a Calculator Use a calculator to find the values of a.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  524 CHAPTER 6 ■ Trigonometric Functions: Right Triangle Approach Angles (p. 472) An angle consists of two rays with a common vertex. One of the rays is the initial side, and the other the terminal side. An angle can be viewed as a rotation of the initial side onto the terminal side. If the rotation is counterclockwise, the angle is positive; if the rotation is clockwise, the angle is negative. ¨ terminal side initial side A B O Notation: The angle in the figure can be referred to as angle AOB, or simply as angle O, or as angle u . Angle Measure (p. 472) The radian measure of an angle (abbreviated rad) is the length of the arc that the angle subtends in a circle of radius 1, as shown in the figure. ¨ Radian measure of ¨ 1 The degree measure of an angle is the number of degrees in the angle, where a degree is 1 360 of a complete circle. To convert degrees to radians, multiply by p/ 180. To convert radians to degrees, multiply by 180/ p. Angles in Standard Position (pp. 473, 494) An angle is in standard position if it is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis. ¨ ¨ y x 0 y x 0 Two angles in standard position are coterminal if their sides coincide. The reference angle u associated with an angle u is the acute angle formed by the terminal side of u and the x-axis. Length of an Arc; Area of a Sector (pp. 475–476) Consider a circle of radius r. ¨ r A s The length s of an arc that subtends a central angle of u radi- ans is s  r u . The area A of a sector with central angle of u radians is A  1 2 r 2 u . Circular Motion (pp. 476–477) Suppose a point moves along a circle of radius r and the ray from the center of the circle to the point traverses u radians in time t. Let s  r u be the distance the point travels in time t. The angular speed of the point is v  u/ t . The linear speed of the point is √  s/ t . Linear speed √ and angular speed v are related by the formula √  rv.

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  Trigonometry

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

    (Publisher)

  1.4 Evaluating Trigonometric Functions: Exactly and with Calculators 35 EXAMPLE 2 Evaluating the Trigonometric Functions Exactly for 458 Evaluate the six trigonometric functions for an angle that measures 45°. Solution: Label the sides of the 458-458-908 right triangle with respect to one of the 45° angles. Use the right triangle ratio definitions of the sine, cosine, and tangent functions. sin 45° 5 opposite hypotenuse 5 x !2 x 5 1 !2 5 1 !2 # !2 !2 5 !2 2 cos 45° 5 adjacent hypotenuse 5 x !2 x 5 1 !2 5 1 !2 # !2 !2 5 !2 2 tan 45° 5 opposite adjacent 5 x x 5 1 Use the reciprocal identities to obtain the values of the cosecant, secant, and cotangent functions. csc 45° 5 1 sin 45° 5 1 !2 2 5 2 !2 5 2 !2 # !2 !2 5 !2 sec 45° 5 1 cos 45° 5 1 !2 2 5 2 !2 5 2 !2 # !2 !2 5 !2 cot 45° 5 1 tan 45° 5 1 1 5 1 The six trigonometric functions evaluated for an angle measuring 45° are sin 45° 5 !2 2 cos 45° 5 !2 2 tan 45° 5 1 csc 45° 5 !2 sec 45° 5 !2 cot 45° 5 1 We see that the following cofunction relationships are indeed true: sin 45° 5 cos 45° sec 45° 5 csc 45° tan 45° 5 cot 45° which is expected, since 45° and 45° are complementary angles. x x 45º √ 2x Adjacent Opposite Hypotenuse STUDY TIP sin 458 5 !2 2 is exact, whereas if we evaluate with a calculator, we get an approximation: sin 45° < 0.7071 36 CHAPTER 1 Right Triangle trigonometry The trigonometric function values for the three special angle measures 1308, 458, and 6082 are summarized in the following table: TRIGONOMETRIC FUNCTION VALUES FOR SPECIAL ANGLES (308, 458, AND 608) u SIN u COS u TAN u COT u SEC u CSC u 30° 1 2 !3 2 !3 3 !3 2 !3 3 2 45° !2 2 !2 2 1 1 !2 !2 60° !3 2 1 2 !3 !3 3 2 2 !3 3 It is important to learn the special values in red for the sine and cosine functions. All other values in the table can be found through reciprocals or quotients of these two functions. Remember that the tangent function is the ratio of the sine to cosine functions.

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