Right Triangles

9 Key excerpts on "Right Triangles"

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  Trigonometry

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

    (Publisher)

  1 *Section 1.5, Example 7 and Exercises 72–74 and 79–80. Thousands of years later, we still find applications of right triangle trigonometry today in sports, surveying, navigation,* and engineering. CHAPTER 1 Right Triangle Trigonometry To the ancient Greeks, trigonometry was the study of Right Triangles. Trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) can be defined as right triangle ratios (ratios of the lengths of sides of a right triangle). 1.1 Angles, Degrees, and Triangles • Angles and Degree Measure • Triangles • Special Right Triangles 1.2 Similar Triangles • Finding Angle Measures Using Geometry • Classification of Triangles 1.3 Definition 1 of Trigonometric Functions: Right Triangle Ratios • Trigonometric Functions: Right Triangle Ratios • Cofunctions RIGHT TRIANGLE TRIGONOMETRY 1.4 Evaluating Trigonometric Functions: Exactly and with Calculators • Evaluating Trigonometric Functions Exactly for Special Angle Measures: 30°, 45°, and 60° • Using Calculators to Evaluate (Approximate) Trigonometric Function Values • Representing Partial Degrees: DD or DMS 1.5 Solving Right Triangles • Accuracy and Significant Digits • Solving a Right Triangle Given an Acute Angle Measure and a Side Length • Solving a Right Triangle Given the Lengths of Two Sides VitalyEdush/Getty Images, Inc. LEARNING OBJECTIVES • Understand degree measure. • Learn the conditions that make two triangles similar. • Define the six trigonometric functions as ratios of lengths of the sides of Right Triangles. • Evaluate trigonometric functions exactly and with calculators. • Solve Right Triangles. 2 CHAPTER 1 Right Triangle Trigonometry In This Chapter We will review angles, degree measure, and special Right Triangles. We will discuss the properties of similar triangles. We will use the concept of similar Right Triangles to define the six trigonometric functions as ratios of the lengths of the sides of Right Triangles (right triangle trigonometry).

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  Trigonometry

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

    (Publisher)

  An isosceles triangle has two equal sides (legs) and therefore has two equal angles opposite those legs. The most important triangle that we will discuss in this course is a right triangle. A right triangle is a triangle in which one of the angles is a right angle 1 90° 2 . Since one angle is 90°, the other two angles must be complementary 1sum to 90° 2 so that the sum of all three angles is 180°. The longest side of a right triangle, called the hypotenuse, is opposite the right angle. The other two sides are called the legs of the right triangle. STUDY TIP In this book when we say “equal angles,” this implies “equal angle measures.” Similarly, when we say an angle is x8, this implies that the angle’s measure is x8. Hypotenuse Leg Leg Right triangle: The Pythagorean theorem relates the sides of a right triangle. It is important to note that length (a synonym of distance) is always positive. PYTHAGOREAN THEOREM In any right triangle, the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (legs). a 2 1 b 2 5 c 2 b a c [ CONCEPT CHECK] TRUE OR FALSE The hypotenuse is always longer than each of the legs of a right triangle. ANSWER True ▼ It is important to note that the Pythagorean theorem applies only to Right Triangles. In addition, it does not matter which leg is called a or b as long as the square of the longest side is equal to the sum of the squares of the smaller sides. EXAMPLE 3 Using the Pythagorean Theorem to Find the Side of a Right Triangle Suppose you have a 10-foot ladder and want to reach a height of 8 feet to clean out the gutters on your house. How far from the base of the house should the base of the ladder be? Solution: Label the unknown side as x. Apply the Pythagorean theorem. x 2 1 8 2 5 10 2 Simplify. x 2 1 64 5 100 10 ft ? 8 ft 10 8 x 8 CHAPTER 1 Right Triangle Trigonometry Solve for x.

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  7 Right Triangles and Vectors Vectors and Right Triangles are closely related. Why a chapter on this one shape? Where do we find Right Triangles? Everywhere! Longitude and latitude, up, down, left and right: we often use a coordinate or grid system to identify position. When you use a grid system the horizontal and vertical lines of the grid are, of course, at right angles to each other. With this chapter we begin our study of trigonometry, the branch of mathematics that enables us to solve triangles. The trigonometric functions are introduced here and are used to solve Right Triangles. Other kinds of triangles (i.e., oblique triangles) are discussed in Chapter 15, and other applications of trigonometry are given in Chapters 16, 17, and 18. We’ll build on what we learned about triangles in Chapter 5, particularly the Pythagorean theorem and the fact that the sum of the interior angles of a triangle is 180°. We’ll also be making use of coordinate axes, described in Sec. 5–1. Also introduced in this chapter are vectors, which we’ll continue to study in Chapter 15. ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Convert angles between: degrees, minutes, and seconds; decimal degrees; and radians. • Find the trigonometric functions of an angle. • Find the acute angle that has a given trigonometric function. • Find the missing sides and angles of a right triangle. • Solve practical problems involving the right triangle. • Resolve a vector into components and, conversely, combine components into a resul- tant vector. • Solve practical problems using vectors. The terms right-angled triangle and right triangle mean the same thing. The first term is more common in the United Kingdom, the second here in North America. In this text we will generally use the shorter right triangle.

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  Analytic Trigonometry with Applications

  • Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  The concepts introduced here will be generalized extensively as we progress through the book. ■ ■ Pythagorean Theorem We start our discussion of Right Triangles with the familiar Pythagorean theorem. In a right triangle, the side opposite the right angle is called the hypotenuse and the other two sides are called legs. If a, b, and c are the lengths of the legs and hypotenuse, respectively, then a 2 + b 2 = c 2 a 2 + b 2 = c 2 . PYTHAGOREAN THEOREM a Leg Hypotenuse Leg b c EXAMPLE 1 Area of a Triangle Find the area of the triangle in Figure 1 on the next page (see Appendix C.2). Solution We need both the base (23) and the altitude (h) to find the area of the triangle. First, we use the Pythagorean theorem to find h. Evaluate squared terms Subtract 529 from both sides Simplify Take the square root of both sides = 200 h 2 = 729 - 529 529 + h 2 = 729 23 2 + h 2 = 27 2 1.3 Trigonometric Ratios and Right Triangles 23 Discard negative square root. (Why?) To two significant digits The area of the triangle is Substitute base and altitude Evaluate To two significant digits ■ Matched Problem 1 Find the area of the triangle in Figure 1 if the altitude is 17 feet and the base is unknown. ■ ■ ■ Trigonometric Ratios If u is an acute angle ( ) in a right triangle, then one of the legs of the triangle is referred to as the side opposite angle u and the other leg is referred to as the side adjacent to angle u (see Fig. 2). As before, the hypotenuse is the side oppo- site the right angle. There are six possible ratios of the lengths of these three sides. These ratios are referred to as trigonometric ratios, and because of their impor- tance, each is given a name: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

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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 also see from this same diagram that, if we have a triangle with one side extended, then the exterior angle e is equal to a + b , the sum of the two interior opposite angles. This is shown drawn in on Figure 4.A.13 . 4.A. (d) Triangles with particular shapes Triangles can come in an infinite variety of shapes, but there are two particular types which have specific names. If a triangle has two sides equal then it is called isosceles (originally by the Greeks who were very keen on geometry – ‘iso’ means ‘equal’ and ‘sceles’ means ‘sides’. ‘Trigonometry’ also comes from the Greeks – ‘trigono’ is the Greek word for triangle.) 4.A Trigonometry in right-angled triangles 139 Figure 4.A.12 Figure 4.A.13 The two equal sides give these triangles a line of symmetry, so that one half folds exactly on to the other half, and the pair of angles opposite the equal sides are also equal. The line of symmetry divides the triangle into two equal right-angled triangles. (See Figure 4.A.14(a) .) The little dashes are there to mark the two equal sides. If a triangle has all three sides equal then it is called equilateral . Such a triangle is pictured in Figure 4.A.14(b) . It will have three lines of symmetry as shown, and will fit exactly onto itself three times in a complete turn. Therefore all its angles are equal, and so must be 60° each. All equilateral triangles can nest into each other, in any chosen corner. Some are shown here in Figure 4.A.15 . They are all similar to each other. (‘Similar’ in maths doesn’t just mean ‘more or less the same as’ but ‘an exact scale model of’ so that all the angles remain the same, and the pairs of sides are all in the same proportion.) 4.A. (e) Congruent triangles – what are they, and when? If two triangles are exactly the same size and shape so that they can be fitted onto each other exactly, they are called congruent . In this case, they will obviously have three equal pairs of angles and three equal pairs of sides.

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  212 Chapter 7 ◆ Right Triangles and Vectors B A C b c a FIGURE 7–10 A right triangle. We will usually label a right triangle as shown here. We label the angles with capital letters A, B, and C, with C always the right angle. We label the sides with lowercase letters a, b, and c, with side a opposite angle A, side b opposite angle B, and side c (the hypotenuse) opposite angle C (the right angle). 7–2 Solving a Right Triangle We will soon see that a great number of applications require us to solve a right triangle. It is an essential skill for technical work. Our tools for solving Right Triangles consist of the trigonometric functions just introduced and, from Chapter 6, the Pythagorean theorem and the fact that, for a right triangle, the sum of the two acute angles must be . 110 104 111 112 113 Solving a Right Triangle When One Side and One Angle Are Known To solve a triangle means to find all missing sides and angles (although in most practical problems we need find only one missing side or angle). We can solve any right triangle if (a) one side and one acute angle are known, or (b) two sides are known. To solve a right triangle when one side and one angle are known, 1. Make a sketch, as in Fig. 7–10. 2. Find the missing angle by subtracting the given angle from . 3. Relate the known side to one of the missing sides by one of the trigonometric functions. Solve for the missing side. 90° tan u  side opposite to u side adjacent to u cos u  side adjacent to u hypotenuse Trigonometric Functions sin u  side opposite to u hypotenuse A  B  90° Sum of the Acute Angles a 2  b 2  c 2 Pythagorean Theorem 90° Applications 17. A certain roof has a rise of 9 in a run of 12. What angle does it make with the horizontal? 18. Find the angle in the machined plate shown in Fig. 7–8. 19. Find the angle in the truss shown in Fig. 7–9. 20. Project: The Framing Square: Figure out how to use the framing square to (a) Lay out any angle.

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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)

  You can find the project at www.stewartmath.com. Hulton Archive/Moviepix/Getty Images Copyright 2017 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. SECTION 6.2 ■ Trigonometry of Right Triangles 485 EXAMPLE 4 ■ Solving a Right Triangle Solve triangle ABC, shown in Figure 7. SOLUTION It’s clear that B  60° . From Figure 7 we have sin 30°  a 12 Definition of sine a  12 sin 30° Multiply by 12  12A 1 2 B  6 Evaluate Also from Figure 7 we have cos 30°  b 12 Definition of cosine b  12 cos 30 Multiply by 12  12 a !3 2 b  6 !3 Evaluate Now Try Exercise 37 ■ Figure 8 shows that if we know the hypotenuse r and an acute angle u in a right triangle, then the legs a and b are given by a  r sin u and b  r cos u The ability to solve Right Triangles by using the trigonometric ratios is fundamental to many problems in navigation, surveying, astronomy, and the measurement of dis- tances. The applications we consider in this section always involve Right Triangles, but as we will see in the next three sections, trigonometry is also useful in solving triangles that are not Right Triangles. To discuss the next examples, we need some terminology. If an observer is looking at an object, then the line from the eye of the observer to the object is called the line of sight (Figure 9). If the object being observed is above the horizontal, then the angle between the line of sight and the horizontal is called the angle of elevation. If the object is below the horizontal, then the angle between the line of sight and the horizontal is called the angle of depression.

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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)

  GIVEN: Right with right and right with right as shown in Figure 5.28; and PROVE: PROOF: With right , the hypotenuse of is ; similarly, is the hypotenuse of right . Because , we denote the common length by c ; that is, . Because , we also have . Then Then so that . Hence, by SSS. Our work with the Pythagorean Theorem would be incomplete if we did not address two issues. The first, Pythagorean triples, involves natural (or counting) numbers as possi-ble choices of a, b, and c . The second leads to the classification of triangles according to the lengths of their sides, as found in Theorem 5.4.7 on page 239. PYTHAGOREAN TRIPLES ABC EDF BC DF BC DF BC c 2 a 2 and DF c 2 a 2 a 2 ( BC ) 2 c 2 and a 2 ( DF ) 2 c 2 , which leads to AC EF a AC EF AB DE c AB DE EDF DE AB ABC ∠ C ABC EDF AC EF AB DE ∠ F DEF ∠ C ABC Figure 5.28 A Pythagorean triple is a set of three natural numbers ( a, b, c ) for which . a 2 b 2 c 2 DEFINITION Copyright 2013 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. Three sets of Pythagorean triples encountered in this section are (3, 4, 5), (5, 12, 13), and (8, 15, 17). These combinations of numbers, as lengths of sides, always lead to a right triangle. Natural-number multiples of any of these triples also produce Pythagorean triples. For example, doubling (3, 4, 5) yields (6, 8, 10), which is also a Pythagorean triple; in Figure 5.29, the two triangles are similar by SSS . The Pythagorean triple (3, 4, 5) also leads to the multiples (9, 12, 15), (12, 16, 20), and (15, 20, 25).

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  Math Starters

  5- to 10-Minute Activities Aligned with the Common Core Math Standards, Grades 6-12

  • Judith A. Muschla, Gary Robert Muschla, Erin Muschla(Authors)
  • 2013(Publication Date)
  • Jossey-Bass

    (Publisher)

  When finding the area of any triangle, always be sure that the units of measurement are the same. Area is measured in square units.

  Problem: Find the area of each triangle below.

  (1)	(2)
  (3)	(4)

  5-67 Finding the Area of a Triangle

  Sometimes it is possible to compare areas without knowing any of the lengths. This requires careful analysis of what you already know.

  Problem: In the figure below, . Which of the following triangles has the largest area: Write an explanation of your answer.

  5-68 Definitions of Trigonometric Ratios for Acute Angles of a Right Triangle (G-SRT.6)

  Three trigonometric ratios are the sine, cosine, and tangent. Each is a ratio of a side of a right triangle to another side:

  • The sine of an acute angle of a right triangle is the ratio of the leg opposite the angle to the hypotenuse.
  • The cosine is the ratio of the leg adjacent to the angle to the hypotenuse.
  • The tangent equals the ratio of the leg opposite the angle to the leg adjacent to the angle.

  Problem: If is an acute angle in a right triangle, , , and are the same if the Right Triangles are similar. Do you agree? Explain your reasoning.

  5-69 Using the Sine and Cosine of Complementary Angles (G-SRT.7)

  When working with all trigonometric functions and right angles, you must first select an acute angle and then write the ratio.

  Problem: As part of her homework, Eva's teacher asked the class to write one trigonometric ratio using the figure below.

  Eva said but her friend Chelsea said that . Their teacher looked at their figures and said they were both correct. Explain how this could be.

  5-70 Using Trigonometric Ratios and the Pythagorean Theorem to Solve Problems G (G-SRT.8)

  The Pythagorean theorem can be used to find the length of a side of a right triangle if the lengths of two sides are known.

  However, if the length of a side of a right triangle and an angle are given, trigonometric ratios may be used to find the length of the other side. Whether you use the sine, cosine, or tangent ratio depends on what information is provided.

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