Mathematics
Trigonometric Functions of General Angles
Trigonometric functions of general angles refer to the sine, cosine, tangent, cosecant, secant, and cotangent functions applied to angles beyond the standard 0 to 90 degrees. These functions can be calculated using the unit circle or by extending the definitions of the functions using periodicity and symmetry properties. They are fundamental in solving problems involving periodic phenomena and oscillations.
Written by Perlego with AI-assistance
Related key terms
1 of 5
9 Key excerpts on "Trigonometric Functions of General Angles"
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 - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
Answer 122° Section 4.3 Summary The trigonometric functions are defined in the Cartesian plane for any angle as follows: Let (x, y) be a 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 sine, cosine, and tangent functions are defined as sinθ = y __ r cosθ = x __ r tanθ = y __ x (x ≠ 0) The range of the sine and cosine functions is [−1, 1], whereas the range of the secant and cosecant functions is (−∞, −1] ∪ [1, ∞). Reference angles and reference right triangles can be used to evaluate trigonometric functions for nonacute angles. 402 CHAPTER 4 Trigonometric Functions of Angles Section 4.3 Exercises Skills In Exercises 1–16, the terminal side of an angle θ in standard position passes through the indicated point. Calculate the values of the six trigonometric functions for angle θ. 1. (3, 4) 2. (12, 5) 3. (3, 6) 4. (8, 4) 5. ( 1 _ 2 , 2 _ 5 ) 6. ( 4 _ 7 , 2 _ 3 ) 7. (−2, 4) 8. (−1, 3) 9. (−4, −7) 10. (−9, −5) 11. (− √ _ 2, √ _ 3) 12. (− √ _ 3, √ _ 2) 13. (− √ _ 5, − √ _ 3) 14. (− √ _ 6, − √ _ 5) 15. ( − 10 __ 3 , 4 _ 3 ) 16. ( − 2 _ 9 , − 1 _ 3 ) In Exercises 17–24, calculate the values for the six trigonometric functions of the angle θ given in standard position, if the terminal side of θ lies on the given line. 17. y = 2x x ≥ 0 18. y = 3x x ≥ 0 19. y = 1 _ 2 x x ≥ 0 20. y = 1 _ 2 x x ≤ 0 21. y = − 1 _ 3 x x ≥ 0 22. y = − 1 _ 3 x x ≤ 0 23. 2x + 3y = 0 x ≤ 0 24. 2x + 3y = 0 x ≥ 0 In Exercises 25–34, indicate the quadrant in which the terminal side of θ must lie in order for each of the following to be true. 25. cos θ is positive and sin θ is negative. 26. cos θ is negative and sin θ is positive. 27. tan θ is negative and sin θ is positive. 28. tan θ is positive and cos θ is negative. 29. sec θ and csc θ are both positive. 30. sec θ and csc θ are both negative. 31. cot θ and cos θ are both positive. 32. cot θ and sin θ are both negative.
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- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
We now turn our attention to graphing the other circular functions: tangent, cotangent, secant, and cosecant. We know the graphs of the sine and cosine functions, and we can get the graphs of the other circular functions from the sinusoidal functions. Recall the reciprocal and quotient identities: tan x 5 sin x cos x cot x 5 cos x sin x sec x 5 1 cos x csc x 5 1 sin x Recall that in graphing rational functions, a vertical asymptote corresponds to a denominator equal to zero (as long as the numerator and denominator have no common factors). As you will see in this section, tangent and secant functions have graphs with vertical asymptotes at the x-values where the cosine function is equal to zero, and cotangent and cosecant functions have graphs with vertical asymptotes at the x-values where the sine function is equal to zero. One important difference between the sinusoidal functions, y 5 sin x and y 5 cos x, and the other four trigonometric functions 1 y 5 tan x, y 5 sec x, y 5 csc x, and y 5 cot x 2 is that the sinusoidal functions have defined amplitudes, whereas the other four trigonometric functions do not (since they are unbounded vertically). The Tangent Function Since the tangent function is a quotient that relies on sine and cosine, let us start with a table of values for the quadrantal angles. SKILLS OBJECTIVES ■ ■ Graph basic tangent, cotangent, secant, and cosecant functions. ■ ■ Graph translated tangent, cotangent, secant, and cosecant functions. CONCEPTUAL OBJECTIVES ■ ■ Understand the relationships between the graphs of cosine and secant functions and the sine and cosecant functions. ■ ■ Understand that graph-shifting techniques for tangent and cotangent are consistent with translations used for sinusoidal functions; but for secant and cosecant functions, we first graph the horizontally translated sinusoidal functions and then we shift up or down depending on the vertical translations.
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
(Philadelphia: Saunders College Publishing, 1990), pp. 236–237; D. E. Smith, History of Mathematics, Vol. II (New York: Dover Publications, Inc., 1953), pp. 614–622. y x y x y x radians _ radians _π radians π radians 3 2π 3 2π Figure 3 Examples of angles in standard position. Name of Function Abbreviation cosine cos sine sin tangent tan secant sec cosecant csc cotangent cot Copyright 201 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. To define the trigonometric functions, we begin by placing the angle u in standard position and drawing in the unit circle x 2 y 2 1, as shown in Figure 4. (Recall from Chapter 1 that the equation x 2 y 2 1 represents the circle of radius 1, with center at the origin.) Notice the notation P ( x , y ) in Figure 4; this stands for the point P , with coordinates ( x , y ), where the terminal side of angle u intersects the unit circle. With this notation, we define the six trigonometric functions of u as follows. 7.2 Trigonometric Functions of Angles 483 x 2 +y 2 =1 y x P(x, y) ¨ Figure 4 P ( x , y ) denotes the point where the terminal side of angle u intersects the unit circle. Definition Trigonometric Functions of Angles Much of our subsequent work in trigonometry will be devoted to exploring the consequences of these definitions. Two initial observations that will help you in memorizing the definitions are these: 1. cos u is the first coordinate of the point where the terminal side of angle u inter-sects the unit circle; sin u is the second coordinate. (You can remember this by noting that, alphabetically, cosine comes before sine.) 2.
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. 6.3 Trigonometric Functions of Any Angle 451 GO DIGITAL Trigonometric Functions of Real Numbers To define a trigonometric function of a real number (rather than an angle), let t represent any real number. Then imagine that the real number line is wrapped around a unit circle, as shown in the figures below. (Recall from Section 1.1 that a unit circle has a radius of 1.) Note that positive numbers correspond to a counterclockwise wrapping and negative numbers correspond to a clockwise wrapping. (1, 0) ( , ) x y t t t > 0 θ y x (1, 0) ( , ) x y t t t < 0 θ y x As the real number line wraps around the unit circle, each real number t corresponds to a central angle θ (in standard position). Moreover, the circle has a radius of 1, so the arc intercepted by the angle θ will have a (directional) length of t. This means that if θ is measured in radians, then t = θ . So, you can define sin t as sin t = sin(t radians), cos t = cos(t radians), tan t = tan(t radians), and so on. Furthermore, each t-value corresponds to a point (x, y) on the unit circle, so sin t = y 1 = y, cos t = x 1 , and tan t = y x . EXAMPLE 8 Evaluating Trigonometric Functions a. Evaluate f (t) = sin t for t = 1 and t = 7π 2. b. Evaluate f (t) = cos t for t = -2π3, which corresponds to the point ( -1 2, -√3 2) on the unit circle. Solution a. Using a calculator in radian mode, f (1) = sin 1 ≈ 0.8415. The angles t = 7π 2 and t = 3π 2 are coterminal, so f ( 7π 2 ) = sin 7π 2 = sin 3π 2 = -1. b. Using the point (-1 2, -√3 2), it follows that f ( - 2π 3 ) = cos ( - 2π 3 ) = x = - 1 2 . Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com a.
eBook - PDF- Geoffrey Berresford, Andrew Rockett(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
524 Chapter 8 Trigonometric Functions 1960 320 340 360 380 400 1970 1980 1990 2000 2010 Year CO 2 concentration (parts per million) Source: National Oceanic and Atmospheric Administration Introduction The graph on the previous page shows the annual cycle of temperatures, varying “periodically” from winter cold to summer heat and back to winter cold. In this chapter we will study trigonometric functions and use them to model this and other periodic behavior, from business cycles to home heating costs. It is assumed that you have studied trigonometry at some time in the past. The first two sections review trigonometry from the beginning, but selectively rather than comprehens-ively, covering only the topics that will be needed for applications. Later sections discuss differentiation and integration of trigonometric functions. We begin with triangles, angles, and radian measure. Right Triangles A triangle with a right (90°) angle is called a right triangle. The lengths of the sides of a right triangle are related by the Pythagorean Theorem.* 8.1 Triangles, Angles, and Radian Measure Pythagorean Theorem In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. a 2 1 b 2 5 c 2 a b hypotenuse c H11005 a 2 H11001 b 2 *This theorem was first attributed to the school of the Greek philosopher Pythagoras (sixth century b.c.), although the Babylonian tablet “Plimpton 322” shows that it was known several centuries earlier. Copyright 2016 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.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
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. tan θ = 2.752 Solution (a) Use a calculator to evaluate the inverse cosine function. Round to the nearest degree. θ ≈ 29° Solution (b) Use a calculator to evaluate the inverse tangent function. θ = tan −1 (2.752) ≈ 70.03026784° Round to the nearest degree. θ ≈ 70° Your Turn Use a calculator to find θ given sin θ = 0.7739. Round the answer to the nearest degree. Answer 51° θ = cos −1 (0.8734) ≈ 29.14382196° 536 CHAPTER 6 Trigonometric Functions EXAMPLE 4 Solving a Right Triangle Given Two Sides Solve the right triangle—find a, α, and β. Solution STEP 1 Determine accuracy. The given sides have four significant digits; therefore, round final calculated values to four significant digits. STEP 2 Solve for α. The cosine of an angle is equal to the adjacent side over the hypotenuse. Evaluate the right side with a calculator. cos α ≈ 0.528621338 Write the angle α in terms of the inverse cosine function. α ≈ cos −1 (0.528621338) Use a calculator to evaluate the inverse cosine function. α ≈ 58.08764855° Round α to four significant digits. α ≈ 58.09° STEP 3 Solve for β. The two acute angles in a right triangle are complementary. α + β = 90° Substitute α ≈ 58.09°. 58.09 + β ≈ 90° Solve for β. β ≈ 31.91° The answer is already rounded to four significant digits. STEP 4 Solve for a. Use the Pythagorean theorem since the lengths of two sides are given. a 2 + b 2 = c 2 Substitute given values for b and c. a 2 + 19.67 2 = 37.21 2 Solve for a. a ≈ 31.5859969 Round a to four significant digits. a ≈ 31.59 cm STEP 5 Check the solution. Angles are rounded to the nearest hundredth degree, and sides are rounded to four significant digits of accuracy.
- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
Chapter 2 Trigonometric Functions 53 (B) Since x 4 = sin 45° and sin 45° = 1 2 ; x 4 = 1 2 ; x = 4 2 Since y 4 = cos 45° and cos 45° = 1 2 ; y 4 = 1 2 ; y = 4 2 (C) Since 5 x = sin 60° and sin 60° = 3 2 ; 5 x = 3 2 ; x = 10 3 Since 5 y = tan 60° and tan 60° = 3 ; 5 y = 3 ; y = 5 3 89. (A) Identity (4) (B) Identity (9) (C) Identity (2) 91. 2 93. Since for all x in the domain of f (that is, all real numbers) 5 sin(2x + 2) = 5 sin(2x), f is periodic. Since 5 sin(2x + 2) = 5 sin 2(x + 1) = f(x + 1), the period p of this function is 1. 95. There is no number p such that sin( x + p) x + p = sin x x for all x 0 (the domain of h). 97. S 1 = 1 S 2 = S 1 + cos S 1 = 1 + cos 1 = 1.540302 S 3 = S 2 + cos S 2 = 1.540302 + cos 1.540302 = 1.570792 S 4 = S 3 + cos S 3 = 1.570792 + cos 1.570792 = 1.570796 S 5 = S 4 + cos S 4 = 1.570796 + cos 1.570796 = 1.570796 2 = 1.570796 CHAPTER 2 REVIEW EXERCISE 1. (A) r = rad 180° d = 180 60 = 3 (B) r = rad 180° d = 180 45 = 4 (C) r = rad 180° d = 180 90 = 2 2. (A) d = 180° rad r = 180 · 6 = 30° (B) d = 180° rad r = 180 · 2 = 90° (C) d = 180° rad r = 180 · 4 = 45° 3. A central angle of radian measure 2 is an angle subtended by an arc with length twice the length of the radius of the circle. 4. An angle of radian measure 1.5 is larger, since the corresponding degree measure of the angle would be 1.5 180° or, approximately, 85.94°. Chapter 2 Review Exercise 54 5. (A) d = 180° rad r = 180 · 15.26 = 874.3° (B) r = rad 180° d = 180 (–389.2) = –6.793 rad 6. V = r = 25(7.4) = 185 ft/min 7. = V r = 415 5.2 = 80 rad/hr 8. Let Q(a, b) be the point on ray OP that lies on the unit circle (see figure). Segment OP has length r = (–4) 2 + 3 2 = 25 = 5. The coordinates of Q(a, b) are obtained by dividing the coordinates of P by r = 5: a = – 4 5 and b = 3 5 . Apply the definition of the trigonometric functions to Q – 4 5 , 3 5 .
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
