ࡱ>  %` bjbjNN 4,,Ghhhhhhh|@1@1@1@1,2|p@2.*6*6*6*6*6*6*63>5>5>5>5>5>5>$"BhDY>h7*6*677Y>hh*6*6*@<<<7h*6h*63><73><<hh<*62 jJ;@1K9J<<@@0p@<E;pE<<\Ehk<,*6v6T<6D87}*6*6*6Y>Y>< *6*6*6p@7777|||,@1|||@1|||hhhhhh 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:  INCLUDEPICTURE "http://home.alltel.net/okrebs/Chptr2-1.gif" \* MERGEFORMATINET   HYPERLINK "http://www.ies.co.jp/math/products/trig/applets/sinBox/sinBox.html" Click for demo of Sine Function (Manipula Math)  HYPERLINK "http://www.ies.co.jp/math/products/trig/applets/cosbox/cosbox.html" Click for demo of Cosine Function (Manipula Math) Let the point P(x,y) be a point on the circle x2 + y2 = r2 and 0 is an angle in standard position. We define the following: Sin q = y/r Cos q = 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 q in standard position goes through (-2, -5), find the Sin qand Cos q. First, draw a sketch: INCLUDEPICTURE "http://home.alltel.net/okrebs/Chptr2-2.gif" \* MERGEFORMATINET  Calculate r: (-2)2 + (-5)2 = 29 = r2 Thus, INCLUDEPICTURE "http://home.alltel.net/okrebs/Chptr2-3a.gif" \* MERGEFORMATINET  Thus INCLUDEPICTURE "http://home.alltel.net/okrebs/Chptr2-4.gif" \* MERGEFORMATINET   2) If theta is a second quadrant angle and sin q= 12/13, find Cos q. 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 r = 13. Find x by using x2 + y2 = r2 x2 + 144 = 169 x2 = 25 x = 5 or -5. Take -5 Must be in second quadrant, remember? Thus, Cos q= -5/13  Signs of the Sine and Cosine Functions Study the following table for the correct signs: FunctionQuad IQuad IIQuad IIIQuad IV Sine + + - - Cosine + - - +  Quadrantal points 1) Find the Sin 90o and Cos 90o Solution: The terminal side of a 90o angle is on the y-axis (0, y) x = 0, y = y and r = y Thus, the Sin 90o = y/y = 1 and Cos 90o = 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. 2) Find the Sin 180o and Cos 180o Solution: The terminal side of a 180o angle is on the negative x-axis. Choose the point (-1, 0) (See note above) x = -1, y = 0, r = 1 Thus, Sin 180o = 0/1 = 0 and Cos 180o = -1/1 = -1 3) Find the Sin 540o and Cos 540o Solution: Since the angle 540o has the same terminal side as 180o, the Sine and Cosine functions have the same value as problem # 2. This leads to the conclusion that the trig functions repeat their value every 360o or 2p Conclusion: Sin (q+ 360o) = Sin q Cos (q+ 360o) = Cos q Sin ( q+ 2p) = Sin q Cos ( q+ 2p) = Cos q Evaluating and Graphing Sine and Cosine Sines and Cosines of Special Angles 30o, 45o and 60o 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: INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-1.gif" \* MERGEFORMATINET  :2 Look at the triangle below:  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-2.gif" \* MERGEFORMATINET  Sin 30o = y/r = 1/2, while the Cos 30o = x/r = INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-3.gif" \* MERGEFORMATINET  Sin 60o = y/r = INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-3.gif" \* MERGEFORMATINET , while the Cos 60o = x/r = 1/2 In a 45-45-90 triangle, the sides are in ratio of 1 : 1 : INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-4.gif" \* MERGEFORMATINET  Study the triangle below:  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-5.gif" \* MERGEFORMATINET  Sin 45o = x/r = INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-6.gif" \* MERGEFORMATINET , while Cos 45o = INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-6.gif" \* MERGEFORMATINET   The wise person will memorize the following chart: Degrees radians Sin q Cos q000130p/61/2 INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-3.gif" \* MERGEFORMATINET 45p/4 INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-6.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-6.gif" \* MERGEFORMATINET 60p/3 INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-3.gif" \* MERGEFORMATINET 1/290p/210 The graph of Sine and Cosine Functions y = Sin x  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-7.gif" \* MERGEFORMATINET   HYPERLINK "http://www.ies.co.jp/math/products/trig/applets/graphSinX/graphSinX.html" Demonstration of Sine Graph (Manipula Math) Notice that this graph is a periodic graph. It repeats the same graph every 2punits. 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  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-8.gif" \* MERGEFORMATINET   HYPERLINK "http://www.ies.co.jp/math/products/trig/applets/graphCosX/graphCosX.html" Demonstration of Cosine Graph (Manipula Math) This graph is similar to the previous shape. It is also a periodic graph with the cycle being 2p. 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.  INCLUDEPICTURE "http://home.alltel.net/okrebs/C3-20.gif" \* MERGEFORMATINET  To find the reference angle a, simply use the chart above to locate the angle q. Example: If q =120,then you are in quadrant II. Thus, use the formula 180 - 120 to get a reference angle of 60. Example: If q = 195, then you are in quadrant III. Thus, use the formula 195 - 180 to get a reference angle of 15. Example: If q = 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 Sin 135o = Sin ( 180o - 135o) = Sin 45o Cos 310o = Cos (360o - 310o) = Cos 50o Sin 210o = Sin (210o - 180o) = - Sin 30o (Sin is negative in third quad) Cos 112o = Cos (180o - 112o) = - Cos 68o (Cos is negative in 2nd quad)  The Four Other Trig Functions The following are the defintions of the other 4 trig functions tangent of q: tan q = y/x cotangent of q: cot q = x/y secant of q: sec q = r/x cosecant of q : csc q = r/y  These four trig functions can be written in terms of sin and cos of q  INCLUDEPICTURE "http://home.alltel.net/okrebs/D1.gif" \* MERGEFORMATINET  The last one shows that the cotangent and tangent are reciprocal functions. Secant and cosine, as well as Cosecant and sine are reciprocal funtions.   INCLUDEPICTURE "http://home.alltel.net/okrebs/D2.gif" \* MERGEFORMATINET  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:  INCLUDEPICTURE "http://home.alltel.net/okrebs/D3.gif" \* MERGEFORMATINET   HYPERLINK "http://www.ies.co.jp/math/products/trig/applets/graphTanX/graphTanX.html" Demonstration of the Tangent Graph (Manipula Math) Period length is p zeros at 0, p, 2p undefined at p/ 2, 3p / 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:  INCLUDEPICTURE "http://home.alltel.net/okrebs/D4.gif" \* MERGEFORMATINET  Period length is p zeros are at: p/ 2, 3p / 2 undefined at: 0, p, 2p This again corresponds with the zeros of the sine and cosine, simply reversed from the tangent graph.  Secant graph  INCLUDEPICTURE "http://home.alltel.net/okrebs/D5.gif" \* MERGEFORMATINET  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 2p Keep in mind that when a graph is undefined, there is a vertical asymptote.  Cosecant Graph:  INCLUDEPICTURE "http://home.alltel.net/okrebs/D6.gif" \* MERGEFORMATINET  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 2p  Example problems 1) Find the other trig functions if sin q = 3/5 and q is in quadrant II. y = 3, r = 5, therefore x = -4. Negative because we are in quad II cos q = -4/5 tan q = -3/4 csc q = 5/3 sec q = -5/4 cot q = -4/3 2) Find each of the values for the trig functions using your Ti-82 graphing calculator. Round to 4 significant digits. a) Tan 115o Make sure you are in degree mode. Type tan 115. Answer -2.145 b) Cot 95o 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 Inverse Trig Functions 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.  INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-1.gif" \* MERGEFORMATINET  If we use the domain: -p/2 < x < p/2, 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. y = sin-1 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:  INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-2.gif" \* MERGEFORMATINET  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!! 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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:  INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-3.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-4.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-5.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://home.alltel.net/okrebs/Ch7-5-6.gif" \* MERGEFORMATINET  The domain for the inverse cosine is -1 < x < 1, with the range at 0 < y < p. 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 -p/2 < y < p/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. 1) Cos-1 (-.5) = 2.09 rounded to nearest hundredth. 2) Sin-1(-.75) = -.85 3) Tan -1 (5) = 1.38 Find the answers in degree mode. Set calculator to degree mode. 4) Cos-1 (.8972) = 26.2o 5) Sin-1 (.3333) = 19.5o 6) Tan-1 (3.2) =72.6o  Problems without using calculator 1) Tan-1 (-1) = x means tan x = -1. In the fourth quadrant x = -45o or 315o _ __ 2) Sin-1 ( \/3/2) = x means that Sin x = \/3/2 In the first quadrant this is 60o 3) Tan(Tan-1 (.5)) = x. Since .5 is in the domain of the arctan and these function are inverse operations the answer is .5 4) Cos-1 (Cos 240o) = x Since 240o is not in the range of arccos, we need to do this in two steps. Cos 240o = -.5, thus Cos-1 (-.5) = 120o . Remember, for the inverse cosine, the answer has to come out in the first or second quadrant! 5) Cos(Tan-1 (2/3)) Since 2/3 is positive, the tan q = 2/3 with the angle being in the first quadrant. Thus y = 2 when x =3 which makes r = \/ 13 ___ ___ Thus the cos q = x/r = 3/ \/ 13 = 3 \/ 13 / 13 6) Cos( Sin-1 ( -4/5)) Since the number is negative, the sin q=-4/5 is in the fourth quadrant. Thus y = -4 and r = 5 which makes x = 3 Thus , the cos q = x/r = 3/5  INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  Notice , we could do the last two problems without really knowing the size of the angle!!  Measurement of Angles Definitions  INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  1) Angle - two rays joined at a common point called a vertex point.  INCLUDEPICTURE "http://home.alltel.net/okrebs/Image1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  2) Revolution - a common unit used to measure large angles, like the number of revolutions a car wheel makes traveling at 10 mph.  INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  3) Degree - a common unit used to measure smaller angles. There are 360 degrees in 1 revolution. 1/2 of a revolution = 180 degrees, 1/4 rev = 90o Degrees can be divided into smaller units of minutes and seconds. 1 degree equals 60 minutes, while 1 minute equals 60 seconds. Examples 15.4o = 15o + .4(60)' = 15o 24' 50o30''15" = 50o + (30/60)o + (15/3600)o = 50.5042o  INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  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.  INCLUDEPICTURE "http://home.alltel.net/okrebs/Image2.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://home.alltel.net/okrebs/paw_red.gif" \* MERGEFORMATINET  5) Radian measure - the number of radius units in the length of an arc AB s = r0  INCLUDEPICTURE "http://home.alltel.net/okrebs/Image3.gif" \* MERGEFORMATINET   Changing radians to degrees and degrees to radians. To change degrees to radians, multiply by opqrstuvwƩǩȩɩ˩̩68:<>@Īƪʪ̪ҪԪ֪ڪܪ|~z|~ڮڮڮڮڮڮ|ڮjh$KU+h$K56B* CJ$OJQJ\]aJ$phf&h$K56>*B* CJ$\]aJ$phf#h$K56B* CJ$\]aJ$phfjzh$KB*Uphf3jh$KB*Uphf3h$Kh$KB*phf3jh$KB*Uphf3j(h$KB*Uphf30­"#*,XY`bopxzƮǮήЮ߮57>@\^߯PQSTVacnp²²²h$KB*phf3jAh$KU&h$K56B* CJ$H*\]aJ$phf&h$K56B* CJH*\]aJphf#h$K56B* CJ$\]aJ$phfh$KF¶öĶŶƶǶȶɶʶ˶̶Ͷζ϶жѶҶӶԶնֶgd$Kgd$K"#lm} pr.2ڳ޳ "fjlp$ Z[xx,jh$K56B* CJ$U\]aJ$phf&h$K56>*B* CJ$\]aJ$phf+h$K56B* CJ$OJQJ\]aJ$phfh$Kh$KB*phf3&h$K56B* CJH*\]aJphf&h$K56B* CJ$H*\]aJ$phf#h$K56B* CJ$\]aJ$phf-[\]bֶHIJK8vnnfnTTnnLnnjh$KU#h$K56B* CJ$\]aJ$ph3jDh$KUjh$KU#h$K56B* CJ0\]aJ0ph3#h$K56B*CJ0\]aJ0phf3jh$KUh$Kh$KB*phf3#h$K56B* CJ$\]aJ$ph#h$K56B* CJ$\]aJ$phf,jh$K56B* CJ$U\]aJ$phf,jh$K56B* CJ$U\]aJ$phfֶ¸/8XZ\QX RVgd$K89:;=¸ø./78<=BCRSWXZ[ghrsܺݺ޺ߺYZ[\]յյյյյյյյյjh$KUjh$KUj"h$KU&h$K56B* CJ$H*\]aJ$ph3jWh$KUh$K#h$K56B* CJ$\]aJ$ph3h$KCJ$aJ$jh$KUjh$KU>PQVWXY &(>@黳us]uuIu]u]&h$K56B* CJ$H*\]aJ$ph+h$K56B* CJ$OJQJ\]aJ$phU#h$K56B* CJ$\]aJ$ph#h$K56B*CJ$\]aJ$phf#h$K56B*CJ0\]aJ0phfjh$KUjh$KUjh$KU&h$K567B* CJ0\]aJ0ph3#h$K56B* CJ0\]aJ0ph3h$K#h$K56B* CJ$\]aJ$ph3p/180 310o = 310 x p /180 = 31 p/18 rads To change radians to degrees, multiply by 180/p 3p = 3px 180/p = 540o 5 rads = 5 x 180/p =286.5o  Angles in the co-ordinate system An angle in the co-ordinate system is usually placed in standard position. 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