Mathematics
Trigonometric Ratios
Trigonometric ratios are relationships between the angles and sides of a right-angled triangle. The three primary trigonometric ratios are sine, cosine, and tangent, which are defined as the ratios of the lengths of the sides of the triangle. These ratios are fundamental in trigonometry and are used to solve for unknown angles or sides in a triangle.
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11 Key excerpts on "Trigonometric Ratios"
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
OBJECTIVES ◆ Write the Trigonometric Ratios for the sine, cosine, and tangent of an angle using the basic terms of a right triangle. ◆ Find the value of a trigonometric ratio using a scientific calculator. ◆ Use a trigonometric ratio to find angles. ◆ Use a trigonometric ratio to find sides. ◆ Solve a right triangle. ◆ Solve application problems involving Trigonometric Ratios and right triangles. Computer Support Specialist Computer technician reparing a computer CHAPTER 13 Right Triangle Trigonometry CandyBox Images/Shutterstock.com 424 CHAPTER 13 ◆ Right Triangle Trigonometry Trigonometric Ratios 13.1 Many applications in science and technology require the use of triangles and trigonometry. Early applications of trigonometry, beginning in the second century b.c., were in astronomy, surveying, and navigation. Applications that you may study include electronics, the motion of projectiles, light refraction in optics, and sound. In this chapter, we consider only right triangles. A right triangle has one right angle, two acute angles, a hypotenuse, and two legs. The right angle, as shown in Figure 13.1, is usually labeled with the capital letter C . The vertices of the two acute angles are usually la- beled with the capital letters A and B . The hypotenuse is the side opposite the right angle, the longest side of a right triangle, and is usually labeled with the lowercase letter c . The legs are the sides opposite the acute angles. The leg (side) opposite angle A is labeled a , and the leg opposite angle B is labeled b . Note that each side of the triangle is labeled with the lowercase of the letter of the angle opposite that side. The two legs are also named as the side opposite angle A and the side adjacent to (or next to) angle A or as the side opposite angle B and the side adjacent to angle B .
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
• Define the six trigonometric functions as ratios of lengths of the sides of right triangles. • Solve right triangles. • Define the six trigonometric functions as ratios of x- and y-coordinates and distances in the Cartesian plane. • Evaluate trigonometric functions for nonacute angles. • Relate degree and radian measures. • Draw the unit circle and label the sine and cosine values for special angles (in both degrees and radians). • Graph sine and cosine functions (amplitude, period, and translations). • Graph tangent, cotangent, secant, and cosecant functions. In This Chapter We will define the trigonometric functions in three ways (all of which are consistent with each other).We will solve right triangles using right triangle trigonometry. We will introduce radian measure, which allows us to formulate trigonometric functions of real numbers. We will graph the trigonometric functions, and we will solve applications involving trigonometric functions. 6.1 Angles, Degrees, and Triangles 497 6.1 Angles, Degrees, and Triangles SKILLS OBJECTIVES • Find the complement and supplement of an angle. • Use the Pythagorean theorem to solve a right triangle. • Solve 30°-60°-90° and 45°-45°-90° triangles. • Use similarity to determine the length of a side of a triangle. CONCEPTUAL OBJECTIVES • Understand that degrees are a measure of an angle. • Understand that the Pythagorean theorem applies only to right triangles. • Understand that to solve a triangle means to find all of the angle measures and side lengths. • Understand the difference between similar and congruent triangles. 6.1.1 Angles and Degree Measure 6.1.1 Skill Find the complement and supplement of an angle. 6.1.1 Conceptual Understand that degrees are a measure of an angle. The study of trigonometry relies heavily on the concept of angles. Before we define angles, let us review some basic terminology. A line is the straight path connecting two points (A and B) and extending beyond the points in both directions.
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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. It is essential that you develop the ability to analyze a prob-lem in order to determine the mathematical principles that are involved in the solution. The solution is done in orderly steps based on mathematical facts. When solving a problem, it is important that you understand the trigonometric operations involved rather than mechanically plugging in values. To solve more complex problems, such as those found later in this section, an understanding of the principles involved is essential. 33–1 RATIO OF RIGHT TRIANGLE SIDES In a right triangle, the ratio of two sides of the triangle determines the sizes of the angles, and the angles determine the ratio of the sides. For example, in Figure 33–1, the size of angle A is determined by the ratio of side a to side b . When side a 5 1 inch and side b 5 2 inches, the ratio of a to b is 1:2. If side a is increased to 2 inches and side b remains 2 inches, as shown in Figure 33–2, the ratio of a to b is 1:1. Figure 33–3 compares the two ratios and shows the change in angle A . UNIT 33 INTRODUCTION TO TRIGONOMETRIC FUNCTIONS 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. Unit 33 INTRODUCTION TO TRIGONOMETRIC FUNCTIONS 839 A B C a 5 1 in.
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
2 Trigonometric Functions IN THIS PART . . . Define the basic trig functions using the lengths of the sides of a right triangle. Determine the relationships between the trig cofunctions and their shared sides. Extend your scope to angles greater than 90 degrees using the unit circle. Investigate the ins and outs of the domains and ranges of the six trig functions. Use reference angles to compute trig functions. Apply trig functions to real-world problems. CHAPTER 6 Describing Trig Functions 91 Chapter 6 Describing Trig Functions B y taking the lengths of the sides of right triangles or the chords of circles and creating ratios with those numbers and variables, our ancestors initi- ated the birth of trigonometric functions. These functions are of infinite value, because they allow you to use the stars to navigate and to build bridges that won’t fall. If you’re not into navigating a boat or engineering, then you can use the trig functions at home to plan that new addition. And they’re a staple for students going into calculus. You may be asking, “What is a function? What does it have to do with trigonom- etry?” In mathematics, a function is a mechanism that takes a value you input into it and churns out an answer, called the output. A function is connected to rules involving mathematical operations or processes. The six trig functions require one thing of you — inputting an angle measure — and then they output a number. These outputs are always real numbers, from infinitely small to infinitely large and everything in between. The results you get depend on which function you use. Although in earlier times, some of the function computations were rather tedious, today’s hand-held calculators, and even phones, make everything much easier. IN THIS CHAPTER » Understanding the three basic trig functions » Building on the basics: The reciprocal functions » Recognizing the angles that give the cleanest trig results » Determining the exact values of functions
eBook - PDFPre-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 .
eBook - PDF- 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 .
eBook - PDF- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
To help make all of this clear, look at the ratios for this triangle: Compare the ratios for angle with the ratios for angle u in the definition: If u and are the acute angles in a right triangle, then they are complemen- tary angles and satisfy (Why?)* Substituting for gives us the complementary relationships in the following box. f 90° - u u + f = 90°. f cot u = tan f csc u = sec f cos u = sin f tan u = cot f sec u = csc f sin u = cos f f cot f = x y = Adj Opp tan f = y x = Opp Adj sec f = z x = Hyp Adj cos f = x z = Adj Hyp csc f = z y = Hyp Opp sin f = y z = Opp Hyp u u a a b b c c f Adj = x Hyp = z Opp = y u * Since the sum of the measures of all three angles in a triangle is 180°, and a right triangle has one 90° angle, the two remaining acute angles must have measures that sum to Therefore, the two acute angles in a right triangle are always complementary. 180° - 90° = 90°. For a given acute angle u in a right triangle, use Euclid’s theorem (from Section 1.2) to explain why the value of any of the six Trigonometric Ratios for that angle is independent of the size of the triangle. 1.3 Trigonometric Ratios and Right Triangles 25 Note that the sine of u is the same as the cosine of the complement of u (which is in the triangle shown), the tangent of u is the cotangent of the comple- ment of u, and the secant of u is the cosecant of the complement of u. The trigono- metric ratios cosine, cotangent, and cosecant are sometimes referred to as the cofunctions of sine, tangent, and secant, respectively. ■ ■ Calculator Evaluation For the Trigonometric Ratios to be useful in solving right triangle problems, we must be able to find each for any acute angle. Scientific and graphing calculators can approximate (almost instantly) these ratios to eight or ten significant digits. Scientific and graphing calculators generally use different sequences of steps. Consult the user’s manual for your particular calculator.
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
NOTE: In Example 5, you can determine the height of the building by entering the expression on your calculator. There are a total of six Trigonometric Ratios. We define the remaining ratios for completeness; however, we will be able to solve all application problems in this chapter by using only the sine, cosine, and tangent ratios. The remaining ratios are the cotangent (abbreviated “cot”), secant (abbreviated “sec”), and cosecant (abbreviated “csc”). These are defined in terms of the right triangle shown in Figure 11.33. 40 tan 47 40 tan 33 ( x y ) x y 43 26 69. y 26 x 43 x 40 tan 47 and y 40 tan 33 tan 47 x 40 and tan 33 y 40 x y tan 36 b 6.7 tan b b a cos b 6.9 9.2 cos b a c tan a 7.2 4.5 tan a a b sin 57 a 12 sin a a c b 36 a 6.7 c 9.2 a 6.9 b b 4.5 a 7.2 a c 12 a 57 11.3 ■ The Tangent Ratio and Other Ratios 507 Unless otherwise noted, all content on this page is © Cengage Learning. a b c Figure 11.31 csc a hypotenuse opposite sec a hypotenuse adjacent cot a adjacent opposite 40 ft x y 47° 33° Opposite Adjacent Hypotenuse Figure 11.32 Figure 11.33 T echnology Exploration If you have a graphing calculator, show that tan 23 equals . The identity is true as long as . cos a 0 tan a sin a cos a sin 23 cos 23 For the fraction (where ), the reciprocal is . It is easy to see that cot is the reciprocal of tan , sec is the reciprocal of cos , and csc is the reciprocal of sin . a a a a a a b a ( a 0) b 0 a b 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. EXAMPLE 6 Use the given information to find the missing ratio.
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
542 chapter 9 Trigonometric Functions 9.4 More Trigonometric Functions learning objectives By the end of this section you should be able to evaluate the tangent of any multiple of 30 ◦ or 45 ◦ ( π 6 radians or π 4 radians); find the equation of the line making a given angle with the positive horizontal axis and containing a given point; sketch a radius of the unit circle corresponding to a given value of the tangent function; compute cos θ, sin θ, and tan θ if given just one of these quantities and the location of the corresponding radius; evaluate sec θ, csc θ, and cot θ as 1 divided by the value of one of the other trigonometric functions. Section 9.3 introduced the cosine and the sine, the two most important trigonometric functions. This section introduces the tangent, another key trigonometric function, along with three more trigonometric functions. Definition of Tangent Recall that cos θ and sin θ are defined to be the first and second coordinates of the endpoint of the radius of the unit circle corresponding to θ. The ratio of these two numbers, with the cosine in the denominator, turns out to be sufficiently useful to deserve its own name. Tangent The tangent of an angle θ, denoted tan θ, is defined by tan θ = sin θ cos θ provided that cos θ = 0. The radius of the unit circle corresponding to θ has its initial point at (0, 0) and its endpoint at (cos θ, sin θ). Thus the slope of this line segment Recall that the slope of the line segment connecting (x 1 , y 1 ) and (x 2 , y 2 ) is y 2 -y 1 x 2 -x 1 . equals sin θ-0 cos θ-0 , which equals sin θ cos θ , which equals tan θ. In other words, we have the following interpretation of the tangent of an angle: Tangent as slope tan θ is the slope of the radius of the unit circle corresponding to θ. The following figure illustrates how the cosine, sine, and tangent of an angle are defined: section 9.4 More Trigonometric Functions 543 Θ cos Θ, sin Θ slope tan Θ 1 The radius corresponding to θ has slope tan θ.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
We derived a formula for finding the area of a triangle, given two sides and the included angle. We also proved the Law of Sines and the Law of Cosines for acute triangles. Another unit for measuring angles, the radian, was introduced in the Perspective on Applications section. Key Concepts 11.1 Greek Letters: a, b, g, u • Opposite Side (Leg) • Hypotenuse • Sine Ratio: sin u 5 opposite hypotenuse • Angle of Elevation • Angle of Depression 11.2 Adjacent Side (Leg) • Cosine Ratio: cos u 5 adjacent hypotenuse • Identity: sin 2 u 1 cos 2 u 5 1 11.3 Tangent Ratio: tan u 5 opposite adjacent • Cotangent • Secant • Cosecant • Reciprocal Ratios 11.4 Area of a Triangle: A 5 1 2 bc sin a A 5 1 2 ac sin b A 5 1 2 ab sin g • Law of Sines: sin a a 5 sin b b 5 sin g c • Law of Cosines: c 2 5 a 2 1 b 2 2 2ab cos g b 2 5 a 2 1 c 2 2 2ac cos b a 2 5 b 2 1 c 2 2 2bc cos a or cos g 5 a 2 1 b 2 2 c 2 2ab cos b 5 a 2 1 c 2 2 b 2 2ac cos a 5 b 2 1 c 2 2 a 2 2bc Summary In Review Exercises 1 to 4, state the ratio needed, and use it to find the measure of the indicated line segment to the nearest tenth of a unit. 1. a 16 in. 40° 2. 70° d 8 ft 3. A 80° D B C c 4 in. ABCD 4. f 5 ft Regular pentagon In Review Exercises 5 to 8, state the ratio needed, and use it to find the measure of the indicated angle to the nearest degree. 5. 13 in. a 14 in. 6. A 10 ft u B C 15 ft D 26 ft Isosceles trapezoid ABCD 7. A Rhombus ABCD a 9 cm 12 cm D C B 8. O 24 in. b 7 in. Circle O In Review Exercises 9 to 12, use the Law of Sines or the Law of Cosines to find the indicated length of side or angle mea- sure. Angle measures should be found to the nearest degree; distances should be found to the nearest tenth of a unit. 9. A 49° C B x 57° 8 10. E 14 a 16 15 D F 11. A 40° C B y 20 60° 12. P 14 Q R w 60° 21 In Review Exercises 13 to 17, use the Law of Sines or the Law of Cosines to solve each problem. Angle measures should be found to the nearest degree; distances should be found to the nearest tenth of a unit.
eBook - PDF- 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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