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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 The Trigonometric Functions Chapter 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.1 Trigonometric Functions of Acute Angles Determine the six trigonometric ratios for a given acute angle of a right triangle. Determine the trigonometric function values of 30°, 45°, and 60°. Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle. Given the function values of an acute angle, find the function values of its complement. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Ratios The figure illustrates how a right triangle is labeled with reference to a given acute angle, . The lengths of the sides of the triangle are used to define the six trigonometric ratios: sine (sin) cosecant (csc) cosine (cos) secant (sec) tangent (tan) cotangent (cot) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Sine and Cosine The sine of is the length of the side opposite divided by the length of the hypotenuse: Hypotenuse side opposite sin hypotenuse Side Opposite Side Adjacent to The cosine of is the length of the side adjacent to divided by the length of the hypotenuse. side adjacent to cos hypotenuse Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5 Trigonometric Function Values of an Acute Angle Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 Example Use the triangle shown to calculate the six trigonometric function values of . Solution: opp 24 sin hyp 25 adj 7 cos hyp 25 opp 24 tan adj 7 hyp 25 csc opp 24 hyp 25 sec adj 7 adj 7 cot opp 24 25 24 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7 Slide 8- 7 Reciprocal Functions Reciprocal Relationships 1 csc sin 1 sec cos 1 cot tan Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 Pythagorean Theorem The Pythagorean theorem may be used to find a missing side of a right triangle. 2 2 52 h 2 4 25 h 29 h 2 29 h h 2 5 2 This procedure can be combined with the reciprocal relationships to find the six trigonometric function values. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 Example 5 If the tan , find the other five trigonometric function values of . 2 Solution: Find the length of the hypotenuse. 5 5 29 sin 29 29 2 2 29 cos 29 29 5 tan 2 29 csc 5 29 sec 2 2 cot 5 29 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Slide 8- 10 Function Values of 30 and 60 When the ratio of the opposite side to the hypotenuse is ½, must have a measure of 30. 1 sin 2 2 1 30 Using the Pythagorean theorem the missing side is 3. The missing angle must have a measure of 60. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11 Function Values of 30 and 60 60 2 1 30 3 1 sin 30 2 3 cos30 2 1 3 tan 30 3 3 3 sin 60 2 1 cos 60 2 tan 60 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12 Function Values of 45 The legs of this triangle must be equal, since they are opposite congruent angles. The hypotenuse is found by: 12 12 h 2 11 h 45 2 h 1 2 h2 2h Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 45 1 Slide 8- 13 Function Values of 45 continued 1 2 sin 45 2 2 1 2 cos 45 2 2 1 tan 45 1 1 45 2 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 45 1 Slide 8- 14 Summary of Function Values Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15 Example As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16 Example continued Solution: We begin with a drawing of the situation. We know the measure of an acute angle and the length of its adjacent side. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17 Example continued Since we want to determine the length of the opposite side, we can use the tangent ratio, or the cotangent ratio. opp h tan 30 adj 1.2 1.2 tan 30 h 3 1.2 h 3 0.7 h The balloon is approximately 0.7 mi, or 3696 ft, high. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18 Cofunctions and Complements The trigonometric function values for pairs of angles that are complements have a special relationship. They are called cofunctions. sin cos(90 ) cos sin(90 ) tan cot(90 ) cot tan(90 ) sec csc(90 ) csc sec(90 ) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19 Example Given that sin 40 0.6428, cos 40 0.7660, and tan 40 0.8391, find the six trigonometric function values of 50. 1 csc 40 1.5557 sin 40 1 sec 40 1.3055 cos 40 1 cot 40 1.1918 tan 40 sin 50 cos 40 0.7660 tan 50 cot 40 1.1918 sec50 csc 40 1.5557 cos50 sin 40 0.6428 cot 50 tan 40 0.8391 csc50 sec 40 1.3055 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20 5.2 Applications of Right Triangles Solve right triangles. Solve applied problems involving right triangles and trigonometric functions. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Right Triangles To solve a right triangle means to find the lengths of all sides and the measures of all angles. This can be done using right triangle trigonometry. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22 Example In ABC , find a, b, and B. Solution: a sin 42 16.5 a 16.5sin 42 a 11.0 b cos 42 16.5 b 16.5cos 42 b 12.3 B 16.5 A a 42 b C B = 90 42 = 48 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23 Definitions Angle of elevation: angle between the horizontal and a line of sight above the horizontal. Angle of depression: angle between the horizontal and a line of sight below the horizontal. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24 Example To determine the height of a tree, a forester walks 100 feet from the base of the tree. From this point, he measures the angle of elevation to the top of the tree to be 47. What is the height of the tree? h tan 47 100 h 100 tan 47 h 107.2 ft h 47 100 ft Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25 Bearing Bearing is a method of giving directions. It involves acute angle measurements with reference to a northsouth line. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26 Example An airplane leaves the airport flying at a bearing of N32W for 200 miles and lands. How far west of its starting point is the plane? w 200 w 200sin 32 w 106 w sin 32 200 32 The airplane is approximately 106 miles west of its starting point. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 27 5.3 Trigonometric Functions of Any Angle Find angles that are coterminal with a given angle and find the complement and the supplement of a given angle. Determine the six trigonometric function values for any angle in standard position when the coordinates of a point on the terminal side are given. Find the function values for any angle whose terminal side lies on an axis. Find the function values for an angle whose terminal side makes an angle of 30°, 45°, or 60° with the x-axis. Use a calculator to find function values and angles. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Angle in Standard Position An angle formed by it’s initial side along the positive x-axis, with it’s vertex at the origin, and it’s terminal side placed at the end of the rotation. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 29 Coterminal Angles Two or more angles that have the same terminal side. For example, angles of measure 60 and 420 are coterminal because they share the same terminal side. Example: Find two positive and two negative angles that are coterminal with 30. 30 360 390 30 360 330 30 2(360 ) 750 30 2(360 ) 690 390, 750, 330, and 690 are coterminal with 30. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 30 Trigonometric Functions of Any Angle Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 31 Example Find the six trigonometric function values for the angle shown: Solution: First, determine r. (2,4) r r 2 4 2 2 4 2 r 20 r 45 4 5 r2 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 32 Example continued The six trigonometric functions values are: sin y 4 2 5 r 2 5 5 x 2 5 r 2 5 5 y 4 tan 2 x 2 cos csc r 2 5 5 y 4 2 r 2 5 sec 5 x 2 x 2 1 cot y 4 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 33 Another Example 1 Given that tan , and is in the first quadrant, find 2 the other function values. Solution: Sketch and label the angle. Find any missing sides. (2,1) r r 2 1 2 2 2 r 4 1 r 5 1 sin 1 5 5 5 2 2 5 cos 5 5 1 tan 2 5 5 1 5 sec 2 2 cot 2 1 csc Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 34 Reference Angle The reference angle for an angle is the acute angle formed by the terminal side of the angle and the x-axis. The reference angle can be used when trying to find the trigonometric function values for angles that cover more than one quadrant. (ex. 210) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35 Example Find the sine, cosine, and tangent function values for 210. Solution: Draw the angle. 210 1 3 30 2 Note that there is a 30 angle in the third quadrant. Label the sides of the triangle with 3, 1, and 2 as shown. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 36 Example continued 1 sin 210 2 3 cos 210 2 1 tan 210 2 210 1 3 30 2 Notice that both the sine and cosine are negative because the angle measuring 210 is in the third quadrant. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 37 5.4 Radians, Arc Length, and Angular Speed Find points on the unit circle determined by real numbers. Convert between radian and degree measure; find coterminal, complementary, and supplementary angles. Find the length of an arc of a circle; find the measure of a central angle of a circle. Convert between linear speed and angular speed. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radian Measure An angle measures 1 radian when the angle intercepts an arc on a circle equal to the radius of the circle. 1 radian is approximately 57.3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 39 Converting between Degree and Radian Measure radians 180 To convert from degree to radian measure, multiply by radians 180 180 1. radians . To convert from radian to degree measure, multiply by 180 . radians Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 40 Example Convert each of the following to either radians or degrees. a) 150 b) 75 150 5 150 radians 180 180 6 c) 7 radians 4 7 180 1260 315 4 4 75 5 75 180 180 12 d) 3 radians 3 180 540 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 171.9 Slide 8- 41 Radian Measure Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 42 Example Find the length of an arc of a circle of radius 10 cm associated with an angle of radians. 4 s r s r or s r 10 5 s 10 4 4 2 s 7.85 cm Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 43 Definitions Linear Speed: the distance traveled per unit of time, where s is the distance and t is the time. s v t Angular Speed: the amount of rotation per unit of time, where is the angle of rotation and t is the time. t Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 44 Linear Speed in Terms of Angular Speed The linear speed v of a point a distance r from the center of rotation is given by v = r, where is the angular speed in radians per unit of time. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 45 Example Find the angle of revolution of a point on a circle of diameter 30 in. if the point moves 4 in. per second for 11 seconds. Since t = 11, must be determined before we can solve for . v v r , . r 4in./ sec 0.26 per second 15in. 4 44 t 11 The angle of revolution of 15 15 the point is approximately 2.93 3 radians. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 46 5.5 Circular Functions: Graphs and Properties Given the coordinates of a point on the unit circle, find its reflections across the x-axis, the y-axis, and the origin. Determine the six trigonometric function values for a real number when the coordinates of the point on the unit circle determined by that real number are given. Find function values for any real number using a calculator. Graph the six circular functions and state their properties. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Circular Functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 48 Example Find each of the following function values. 2 3 3 c) tan 4 b) cos a) sin 4 5 d) sec 6 Solutions: 1 3 , 2 2 2 3 a) The coordinates of the point 2 determined by are 1 , 3 3 2 2 2 3 sin y 3 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 49 Example continued b) The coordinates of the point determined by 4 are 2 2 2 , 2 2 cos x 2 4 3 c) The coordinates of the point determined by 4 are 2 2 , tan 3 y 2 2 1 2 2 4 x 2 2 5 d) The coordinates of the point determined by 6 are 3 1 , 2 2 5 1 1 2 3 sec 6 x 3 2 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 50 Graphs of the Sine and Cosine Functions 1. Make a table of values. 2. Plot the points. 3. Connect the points with a smooth curve. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 51 Example 1. Make a table of values. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 52 Example continued 2. Plot the points. 3. Connect the points with a smooth curve. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 53 Domain and Range of Sine and Cosine Functions The domain of the sine and cosine functions is (, ). The range of the sine and cosine functions is [1, 1]. Periodic Function A function f is said to be periodic if there exists a positive constant p such that f(s + p) = f(s) for all s in the domain of f. The smallest such positive number p is called the period of the function. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 54 Amplitude The amplitude of a periodic function is defined as one half of the distance between its maximum and minimum function values. The amplitude is always positive. The amplitude of y = sin x and y = cos x is 1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 55 Graph of y = tan s Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 56 Graph of y = cot s Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 57 Graph of y = csc s Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 58 Graph of y = sec s Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 59 5.6 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin x and y = cos x in the form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D and determine the amplitude, the period, and the phase shift. Graph sums of functions. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Transformed Sine and Cosine Functions: Vertical Translation y = sin x + D and y = cos x + D The constant D translates the graphs D units up if D > 0 or |D| units down if D < 0. Example: Sketch a graph of y = sin x 2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 61 Graphs of Transformed Sine and Cosine Functions: Amplitude y = A sin x and y = A cos x If |A| > 1, then there will be a vertical stretching by a factor of |A|. If |A| < 1, then there will be a vertical shrinking by a factor of |A|. If A < 0, the graph is also reflected across the x-axis. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 62 Example Sketch a graph of y = 3 sin x. The sine graph (y = sin x) is stretched vertically by a factor of 3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 63 Graphs of Transformed Sine and Cosine Functions: Period y = sin Bx and y = cos Bx If |B| < 1, then there will be a horizontal stretching. If |B| > 1, then there will be a horizontal shrinking. If B < 0, the graph is also reflected across the y-axis. 2 . The period will be B Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 64 Example Sketch a graph of y = sin 2x. The sine graph (y = sin x) is shrunk horizontally. 2 The period is 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 65 Graphs of Transformed Sine and Cosine Functions: Horizontal Translation or Phase Shift y = sin (x C) and y = cos (x C) The constant C translates the graph horizontally |C| units to the right if C > 0 and |C| units to the left if C < 0. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 66 Example Sketch the graph of y = sin (x + ). The sine graph (y = sin x) is translated units to the left. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 67 Combined Transformations y = A sin (Bx C) + D and y = A cos (Bx C) + D The amplitude is |A|. The period is The graph is translated vertically D units. The graph is translated horizontally C units. 2 . B Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 68 Example Find the vertical shift, amplitude, period, and phase shift for the following function: y = 2 sin (4x 2) 3. Solution: Write the function in standard form. y 2sin(4 x 2 ) 3 |A| = |2| = 2 means the amplitude is 2 B = 4 means the period 2 is 2 y 2sin 4 x 4 (3) 4 2 C/B = 2 means the phase shift is units to the right. 2 D = 3 means the vertical shift is 3 units down. y 2sin 4 x (3) 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 69 Example continued Sketch y 2sin 4 x (3) 2 Amplitude = 2 Vertical shift = 3 down Phase shift = 2 right Period = First, sketch y = sin 4x. 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 70 Example continued Second, sketch y = 2 sin 4x. Third, sketch y 2sin 4 x 2 . Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 71 Example continued Finally, sketch y 2sin 4 x (3) . 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 72