美国高中数学2-7 Trigonometric Functions

美国高中数学2-7 Trigonometric Functions 7-1 Measurement of Angles Definitions 1) Angle - two rays joined at a common point called a vertex point. Demo: Ferris Wheel (Manipula Math) 2) Revolution - a common unit used to measure large angles, like the number of revolutions a car wheel makes traveling at 10 mph. 3) Degree - a common unit used to measure smaller angles. There are 360 degrees in 1 revolution. 1/2 of oa revolution = 180 degrees, 1/4 rev = 90 Degrees can be divided into smaller units of minutes and seconds. 1 degree equals 60 minutes, while 1 minute equals 60 seconds. Examples ooo15.4 = 15 + .4(60)' = 15 24' ooooo5030''15" = 50 + (30/60) + (15/3600) = 50.5042 4) Radian - the measure of a the central angle when an arc of a circle has the same length as the radius of the circle. 5) Radian measure - the number of radius units in the length of an arc AB s = r0 Changing radians to degrees and degrees to radians. To change degrees to radians, multiply by ,/180 o310 = 310 x , /180 = 31 ,/18 rads To change radians to degrees, multiply by 180/, o,, , ,,x 180/, , ,,, o5 rads = 5 x 180/, ,,,,,, Angles in the co-ordinate system An angle in the co-ordinate system is usually placed in standard position. This means that the vertex is at the origin and its initial ray is along the positive x-axis. A counterclockwise rotation is considered to be positive and a clockwise rotation is considered to be negative. If the terminal side of an angle is standard position lies along an axis, the angle is said to be a qadranutal angle. Two angles in standard postion are called coterminal if they have the same terminal side. Samples 1) Find two angles with the same terminal side, one positive and one negative for each angle. o a) 120 Add 360 to find another positive 120 + o 360 = 480 Subtract 360 to find a negative 120 - o360 = -240 o b) 400 Add or subtract 360 for a positive. 400 - o360 = 40 Subtract enough 360's to make it negative. 400 - 360 - 360 = o-320 7-3 The Sine and Cosine Functions The six trigonometric functions are the 6 different ratios that you can set up from a right triangle. To simplify it, we will form the right triangles with a vertex at the origin and a terminal ray in standard position. Study the following graph: Click for demo of Sine Function (Manipula Math) Click for demo of Cosine Function (Manipula Math) 22 Let the point P(x,y) be a point on the circle x + y = 2r and 0 is an angle in standard position. We define the following: Sin , = y/r Cos , = x/r x and y get their signs from the quadrants they appear in, and r > 0 Example 1) If the terminal side of an angle , in standard position goes through (-2, -5), find the Sin ,and Cos ,. First, draw a sketch: 22 2Calculate r: (-2) + (-5)= 29 = r Thus, Thus 2) If theta is a second quadrant angle and sin ,= 12/13, find Cos ,. Solution: Since the angle is in the second quadrant, x must be negative implying Cos must also be negative. Since the sin is 12/13, this means y = 12 and 222r = 13. Find x by using x + y = r 2x + 144 = 169 2x = 25 x = 5 or -5. Take -5 Must be in second quadrant, remember? Thus, Cos ,= -5/13 Signs of the Sine and Cosine Functions Study the following table for the correct signs: Quad Quad Quad Quad Function I II III IV Sine + + - - Cosine + - - + Quadrantal points oo1) Find the Sin 90 and Cos 90 o Solution: The terminal side of a 90 angle is on the y-axis (0, y) x = 0, y = y and r = y oo Thus, the Sin 90 = y/y = 1 and Cos 90 = 0/y = 0 Note: It doesn't matter what y value I take for this problem. From now on I will choose 1 to make the arithmetic easy. This also goes for points on the x-axis. oo2) Find the Sin 180 and Cos 180 o Solution: The terminal side of a 180 angle is on the negative x-axis. Choose the point (-1, 0) (See note above) x = -1, y = 0, r = 1 oo Thus, Sin 180 = 0/1 = 0 and Cos 180 = -1/1 = -1 oo3) Find the Sin 540 and Cos 540 o Solution: Since the angle 540 has the same oterminal side as 180, the Sine and Cosine functions have the same value as problem # 2. This leads to the conclusion that the trig functions orepeat their value every 360 or 2, Conclusion: oSin (,+ 360) = Sin , oCos (,+ 360) = Cos , Sin ( ,+ 2,) = Sin , Cos ( ,+ 2,) = Cos , 7-4 Evaluating and Graphing Sine and Cosine Sines and Cosines of Special Angles ooo 30, 45 and 60 angles are used many times in mathematics. I strongly urge you to memorize, or at least be able to derive the sine and cosine of these special angles. In a 30-60-90 triangle, the sides are in ratio of 1: :2 Look at the triangle below: oo Sin 30 = y/r = 1/2, while the Cos 30 = x/r = oo Sin 60 = y/r = , while the Cos 60 = x/r = 1/2 In a 45-45-90 triangle, the sides are in ratio of 1 : 1 : Study the triangle below: oo Sin 45 = x/r = , while Cos 45 = The wise person will memorize the following chart: Sin Cos Degrees radians , , 0 0 1 , 30 1/2 ,,, 45 ,,, 60 1/2 ,,, 90 1 0 ,,, The graph of Sine and Cosine Functions y = Sin x Demonstration of Sine Graph (Manipula Math) Notice that this graph is a periodic graph. It repeats the same graph every 2,units. It is increasing from 0 to half pi, decreasing from half pi to negative 1.5 pi and increasing to 2 pi. Then the repeat starts. This matches what happens to the Sine function in the quadrants. Positive in first and second and negative in the third and fourth. Maximum value for the graph is 1 and the minimum value is -1. y = Cos x Demonstration of Cosine Graph (Manipula Math) This graph is similar to the previous shape. It is also a periodic graph with the cycle being 2,. It also matches the signs of the quadrants with quad one being positive, quads two and three, negative and quad 4 back to positive. The difference in these two graphs is the starting point for the Cosine graph. It starts at the maximum value. The Sine curve started at the origin point. An easy way to remember these graphs is to know their 5 important points. The zeros, maximum and minimum points. The Sine curve has zeros at the beginning, middle and end of a cycle. The maximum happens at the 1/4 mark and the minimum appears at the 3/4 mark. The Cosine curve begins and ends with the maximum. It has a minimum at the middle point. Zeros appear at the 1/4 and 3/4 mark of the cycle. Reference Angles All angles can be referenced back to an angle in the first quadrant. This is true because the trig functions are periodic. Study each of the quadrant formulas below to find the reference angles. To find the reference angle ,～ simply use the chart above to locate the angle ,. Example: If , ,,,,～then you are in quadrant II. Thus, use the formula 180 - 120 to get a reference angle of 60. Example: If , , ,？,～ then you are in quadrant III. Thus, use the formula 195 - 180 to get a reference angle of 15. Example: If , = 300, then you are in quadrant IV. Thus, use the formula 360 - 300 to get a reference angle of 60. Relating this idea of reference angles and Sine and Cosine is easy. Determine the reference angle as we did above and put the correct sign on each function. From previous sections the Sine function is positive in quadrants I and II and negative in quadrants III and IV. The Cosine function is positive in quadrants I and IV, while negative in quadrants II and III. Examples ooooSin 135 = Sin ( 180 - 135) = Sin 45 ooooCos 310 = Cos (360 - 310) = Cos 50 ooooSin 210 = Sin (210 - 180) = - Sin 30 (Sin is negative in third quad) ooooCos 112 = Cos (180 - 112) = - Cos 68 (Cos is negative in 2nd quad) Using your TI-82 to evaluate Sine and Cosine 1) To calculate in degrees: Procedure 1 a) Press mode button and highlight degrees. Press enter. b) Type in problem as example: Sin 45. Press enter. c) Answer: .7071067812 Note: Most trig answers are round to 4 decimal places. You can set your calculator to fixed mode by: press mode. Press down arrow and highlight 4. Press enter. Press clear. Now enter the problem: Sin 45 Answer is now .7071 Procedure 2. (Use the degree button) a) Enter problem: Sin 45 b)Press 2nd function. Press "angle" button. You get a menu with the degree symbol as choice 1. Either press 1 or enter. c) Your display should now look like this: Sin o Press enter. 45 Answer is .7071 if you are still in fixed mode!! Note: Procedure 2 works regardless of the mode you are in. 2) To calculate in radian measure. Procedure 1 a) Press mode button. Press down arrow twice. Highlight radian. Press enter. You are now in rads. All problems will now be calculated in rads. b) Enter a problem. Example Cos 5. Press enter. Answer (rounded to 4 decimal places) is .2837 Procedure 2. (Using the rad button) a) Enter a problem such as Sin -2 Press 2nd function. Press angle button. Press down twice or 3. b) Your display should look r Press enter. like: sin -2 Answer is -.9093 Note: The use of the radian button will override the setting in mode. Just like the degree button does. We should be ready to take a look at the other trig functions!! Take me back for some brush-up work!! 7-5 The Four Other Trig Functions The following are the defintions of the other 4 trig functions tangent of ,: tan , = y/x cotangent of ,: cot , = x/y secant of ,: sec , = r/x cosecant of , : csc , = r/y These four trig functions can be written in terms of sin and cos of , The last one shows that the cotangent and tangent are reciprocal functions. Secant and cosine, as well as Cosecant and sine are reciprocal funtions. It is easy to memorize the signs of the six trig functions. All are positive in Quad I, Sine and Csc are positive in quad II, Tan and Cot are positive in Quad III, while Cos and Sec are positive in quad IV. Graphs of the other trig functions: Tangent graph: Demonstration of the Tangent Graph (Manipula Math) Period length is , zeros at 0, ,～ 2, undefined at ,/ 2, 3, / 2 This corresponds to the zeros of sin -- this is where the tangent crosses the x-axis, and to the zeros of the cos -- this is where the tangent is undefined. Cotangent graph: Period length is , zeros are at: ,/ 2, 3, / 2 undefined at: 0, ,～ 2, This again corresponds with the zeros of the sine and cosine, simply reversed from the tangent graph. Secant graph The blue graph is the secant graph. We can generate the secant graph by knowing the graph of the cosine. Remember that they are reciprocal functions. When the cosine is zero, the secant is undefined. When the cosine is at a maximum value, the secant is a minimum. When the cosine is at a minimum, the secant is a maximum. Period length is 2, Keep in mind that when a graph is undefined, there is a vertical asymptote. Cosecant Graph: The blue graph is the cosecant graph. This graph has the same relationship to the sine graph that the cosine and secant graph had. Period length is 2, Example problems 1) Find the other trig functions if sin , = 3/5 and , is in quadrant II. y = 3, r = 5, therefore x = -4. Negative because we are in quad II cos , = -4/5 tan , = -3/4 csc , = 5/3 sec , = -5/4 cot , = -4/3 2) Find each of the values for the trig functions using your Ti-82 graphing calculator. Round to 4 significant digits. o a) Tan 115 Make sure you are in degree mode. Type tan 115. Answer -2.145 o b) Cot 95 Since cot and tan are reciprocals and you don't have a cot button, type it in as: 1/tan 95 -----> Answer -.0875 c) Csc 5 Make sure you are in radian mode. since we don't have a csc button but we remember that csc is the reciprocal of sin, type it in as: 1/sin 5 -----------> Answer: -1.043 d) Sec 11 Since sec and cos are reciprocals, type as: 1/cos 11 ---------> Answer: 226.0 That's it for this section. Let's head into the last section on inverses. Go for it!! I can't go on!! I am completely lost!! Take me back! 7-6 Inverse Trig Functions Try the quiz at the bottom of the page! go to quiz Since the trig functions are all periodic graphs, none of them pass the horizontal line test. Thus, none of the graphs are 1-1 and do not have inverse functions. What we can do is restrict the domain of each of the trig functions to make each one, 1-1. Since the graphs are periodic, if we pick an appropriate domain, we can use all values for the range. If we use the domain: -,/2 < x < ,,,～ we have made the graph 1-1. Notice, every range value is defined if we use this section. The range is: -1 < y < 1 Remember, to find an inverse, it is the reflection about the y = x axis. -1y = sin x is the notation used to represent the inverse sin function. It is also referred to as the arcsin. The graph of the inverse function looks like: Notice, that the range is now the domain and the domain is now the range. Because we have restricted the domain, all answers are now related to the first quadrant or the fourth quadrant. Positive answers in the first and negative answers in the fourth. The inverse function of any of the trig functions will return the angle either measured in degrees or radians. You must be aware that all positive values will return an angle in the first quadrant and negative values will return an answer in the fourth quadrant!! With your calculator set to degree mode: o-1Sin .81 = 54.1 o-1Sin (-.2) = -11.5 ( 348.5) Notice, that domain is: -1 < x < 1. Taking any other value will result in an error message on your calculator. The Cos function and it's inverse and the Tan and it's inverse are also graphed below: The domain for the inverse cosine is -1 < x < 1, with the range at 0 < y < ,, This means that a positive x value will return an answer in the first quadrant and a negative x value will return an answer in the second quadrant. The domain for the arctan is all real numbers with the range -,,, < y < ,/2 The arctan will return the values the same way the inverse sine returns values, in the first and 4th quadrants. Examples for calculator problems Find the answers in radian measure. Set calculator mode to rads. -11) Cos (-.5) = 2.09 rounded to nearest hundredth. -12) Sin(-.75) = -.85 -13) Tan (5) = 1.38 Find the answers in degree mode. Set calculator to degree mode. o-14) Cos (.8972) = 26.2 o-15) Sin (.3333) = 19.5 o-16) Tan (3.2) =72.6 Problems without using calculator -11) Tan (-1) = x means tan x = -1. oo In the fourth quadrant x = -45 or 315 _ __ -12) Sin ( \/3/2) = x means that Sin x = \/3/2 o In the first quadrant this is 60 -13) Tan(Tan (.5)) = x. Since .5 is in the domain of the arctan and these function are inverse operations the answer is .5 o-14) Cos (Cos 240) = x o Since 240 is not in the range of arccos, we o-1need to do this in two steps. Cos 240 = -.5, thus Cos o(-.5) = 120 . Remember, for the inverse cosine, the answer has to come out in the first or second quadrant! -15) Cos(Tan (2/3)) Since 2/3 is positive, the tan , = 2/3 with the angle being in the first quadrant. Thus y = 2 when x =3 which makes r = \/ 13 ___ ___ Thus the cos , = x/r = 3/ \/ 13 = 3 \/ 13 / 13 -1 6) Cos( Sin ( -4/5)) Since the number is negative, the sin ,,,,,, is in the fourth quadrant. Thus y = -4 and r = 5 which makes x = 3 Thus , the cos , = x/r = 3/5 Notice , we could do the last two problems without really knowing the size of the angle!! That's about it for our inverse functions! Are you ready for the sample test? Or would you rather head back and study some more? Current quizaroo # 7 1) Convert 15,/4 to degrees. oa) 675 ob) 60 oc) 180 od) 15 oe) 425 2) Which quadrant or axis is described for sin > 0 and tan < 0 a) I b) II c) III d) IV e) x-axis 3) Find the value of sin 3.4 Round to four decimal places. a) .0593 b) .9982 c) .5592 d) -.2555 e) -.9668 o4) Express in terms of a reference angle: Tan (-105) oa) -tan 75 ob) tan 75 oc) cot 75 od) -cot 75 oe) -tan 105 -1 5) Find the Cot 5.33 rounded to the nearest tenth of a degree. oa) 10.6 ob) .2 oc) 79.4 od) 0 oe) 23.5 click here for answers!! Sample test 1) Convert each degree measure to radians. Leave answers in terms of ,, oooo = b) -210 = c) 160 = d) -120 = a) 250 2) Convert each radian measure to degrees. a) ,,,, , b) ,,,,, , c) ,,,,,, , d) ,,,, , 3) Convert each degree measure to radians. Give answers to the nearest hundredth of a radian. o ooa) 112= b) 252.5 = c) 21035’ = 4) Convert each radian measure to degrees. Give answers to nearest tenth of a degree. Bonus of one point for the section if you also write answers to the nearest ten minutes. a) 1.8 = b) 3.5 = c) 2.41 = 5) Find two angles, one positive and one negative that are coterminal with each given angle. If problem is given in , rads give answer in simpified , rads. ooa) 980 = b) 24025’ = c) ,,,, , 6) Given the point (1, -2) on the terminal side of an angle, find the value of all six trig functions. 7) Express each of the following in terms of a reference angle. ooooa) sin 185 = b) cos 320 = c) tan (-145) = d) cot (-290) = e) sec 3 = f) csc 5 = 8) Find the value of each expression to four significant digits. ooooa) sin 142.5 = b) csc 224 = c) cos 30012’ = d) sec 401 = d) tan 6 = e) cot 7.28 = f) sec 5.1 = g) cot (-1.2) = 9) Find each value to the nearest tenth of a degree. -1-1-1-1 a) Sin.85 = b) Cos(-.28) = c) Cot5.31 = d) Sec20.5 = 10) Find each value to the nearest hundredth of a radian. -1-1-1-1 a) Tan0.56 = b) Cot6.35 = c) Sin.76 = d) Sec3.1 = 11) Find each value to four significant digits. -1-1-1 a) Tan(Cos11/12) = b) Sin(Cos1/5) = c) Csc(Tan1.12) = 12) Fill in the following chart with the exact values that you memorized. Angle Sin , Cos , Tan , Csc , Sec , Cot , o 0 ____ ____ ____ ____ ____ ____ o 30 ____ ____ ____ ____ ____ ____ o 45 ____ ____ ____ ____ ____ ____ o 60 ____ ____ ____ ____ ____ ____ o 90 ____ ____ ____ ____ ____ ____ 13) On graph paper graph each of the following trig functions. a) y = cos x b) y = tan x c) y = csc x This page hosted by Get your own Free Home Page Hopefully, you did well on the sample test and wish to go on!! Or maybe you need to brush up a little more? Sine Box Sine Function The figure shows a definition of sine function. The unit circle is the circle with its center at the origin and a radius of 1. Angle x is formed by rotating OA about the origin to OP. Then the y-coordinate of point P is sin x. Applet Input an angle (degrees), then press enter key or click "Start" button. Activity Using the applet, find angle x that satisfies 1. sin x=0.766 2. sin x=-0.34 3. sin x=0.50 4. sin x=-0.80 Cosine Box Cosine Function The figure shows a definition of cosine function. The unit circle is the circle with its center at the origin and a radius of 1. Angle x is formed by rotating OA about the origin to OP. Then the x-coordinate of point P is cos x. Applet Input an angle(degrees), then press enter key or click "Start" button. Activity Using the applet, find angle x that satisfies 1. cos x=0.766 2. cos x=-0.34 3. cos x=0.50 4. cos x=-0.68