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Chapter Eight Trigonometric Functions The trigonometric functions sine and cosine are also called circular functions since they are defined in terms of distances around a unit circle, a circle with a radius equal to one. Distances are measured around this circle in a counterclockwise direction, starting at the initial point (1, 0). Since the length of the circumference of a circle is the product of and the radius, the distance around the unit circle is circle is a distance of . One-quarter of the way around the , one-half the way around is , and one and a half times around the circle is . If you travel in a clockwise direction around the circle, the distances are negative numbers. The distance from (1, 0) to (0, –1) in the clockwise direction is . How far around the circle do you travel from (0, 1) to (–1, 0): a. in the positive direction b. in the negative direction c. in the positive direction, passing through (1, 0) twice. Ans.: Measurement around the circle is called radian measure. A radian is equal to the length of the radius. For the unit circle, one radian is equal to one. You can mark off a radian around a circle by taking one of the radius lines and wrapping it around the circle. The circumference for any circle is radians, which is for the unit circle. Be sure that you have your calculator in radian mode instead of degree mode before you perform any calculations with the circular functions. , , 136 CHAPTER 8: TRIGONOMETRIC FUNCTIONS The circular functions sine and cosine give the coordinates of the point on the unit circle you reach when you travel around the unit circle a given distance. If the distance you go around the circle from the initial point (1, 0) is x, then the sine (written sin x) is the second coordinate of the point that you reach and the cosine (written cos x) is the first coordinate of that point. You can see several elementary properties of these functions in the unit circle representation above. Properties of Sine and Cosine First, since sin x and cos x are defined for values of x that represent distances traveled on the unit circle, you can see that neither can ever take on a value less than –1 nor greater than 1. No points on the unit circle could ever have coordinates outside these numbers. –1 sin x 1 –1 cos x 1 Start at the initial point (1, 0) and move in the positive (counterclockwise) direction. The values of sin x (the second coordinate of the point on the circle) increase from 0 to 1 as x (the distance along the circle) increases from 0 to to . Then the sine values decrease from 1 to –1 when x changes from , and increase from –1 back to 0 as x goes from to . CHAPTER 8: TRIGONOMETRIC FUNCTIONS 137 That takes you back to the point on the circle where you started, and the sine values repeat through the same numbers as x changes from to , since this takes you around the circle again. Every time you change the value of x by exactly , the value of sin x remains unchanged. The sine function is a cyclic function with a period of , which means that it takes on the same sequence of values in every interval of x-values that is long. The cosine values you obtain as you travel in the positive direction around the circle start at 1 at the initial point, decrease to –1 as x varies from 0 to , then increase from –1 to 1 as x changes from to . The cosine values also repeat after every complete trip around the circle, so the cosine function is also cyclic with a period of . For any integer n: sin (x + n @ ) = sin x cos (x + n @ ) = cos x. You can easily read off some values of sine and cosine from the unit circle. Find: cos , sin sin 0 = 0, cos 0 = 1 sin = 1, cos =0 sin = 0, cos = –1 = –1, cos =0 sin Ans.: 0, 1, –1. Because of the cyclic properties of these functions, you can also find the sine and cosine for any distances around the circle, in either direction, which are integer multiples of sin or : = –1, cos = 1, sin = 0. You can see that the values of cosine are the same for a given distance around the circle, regardless of whether the direction taken is positive or negative. Since the first coordinate of the point on the circle gives the cosine, the value is the same moving clockwise or counterclockwise the same distance. The cosine function is an even function. However, the second coordinates are different for clockwise and counterclockwise paths. The values of the sine function take different signs when you take different directions around the circle. The sine function is an odd function. cos (–x) = cos x and sin (–x) = –sin x , cos . 138 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Moving along the unit circle, you can see another pattern in the values of the circular functions. The values obtained by the sine function are the same as those obtained by the cosine function at the point circle. For example, sin sin = –1 = cos = 1 = cos 0, sin , and sin = 0 = cos units earlier along the = 0 = cos , . Looking at the unit circle, you can see that this is true everywhere around the circle. sin x = cos (x – or cos x = sin (x + ), Note that this means the graph of sin x is a translation of the graph of ) cos x to the right units, or the graph of cos x is a translation of the graph of sin x to the left If you divide the unit circle into fourths, each quarter is called a quadrant of the circle. The first quadrant consists of the points that are between 0 and from the initial point, the second of the points between and , etc. You can see from the unit circle diagram that both sine and cosine are positive in the first quadrant, the sine is positive and the cosine negative in the second, both are negative in the third, and the sine negative and cosine positive in the fourth quadrant. Fitting a right triangle into the first quadrant, and using the Pythagorean Theorem, you can establish the most important trigonometric identity, , for all values of the variable x. units. CHAPTER 8: TRIGONOMETRIC FUNCTIONS When working with triangles, you measure the angles in degrees instead of the radians you use to measure distances around circles. A right angle contains 90°, a straight angle contains 180°, and a circular angle contains 360°. Therefore, when you travel completely around a circle, thedistance you travel is radians, and the size of the angle you have passed through is 360°. Since a distance of 1 radian = or Change to radians: 45°, 60°, 150° Ans.: radians is equivalent to an angle of 360°, degrees , 1 degree = or , . Change to degrees: , and 139 , radians. Among the right triangles that can be fit into the unit circle are a few that you have previously studied. A right triangle having two equal sides has two 45° angles. A 45° angle is equivalent to a distance of radians, or Ans.: 30°, 72°, 315°. th of the way around a circle. Since each of the two sides of the triangle has the same length, say l, the Pythagorean Theorem says that the hypotenuse has a length of = l. The hypotenuse has a length equal to times the length of each other side. In order for such a right triangle to fit into the unit circle, the hypotenuse (the radius of the circle) must be 1. Therefore, each side must have length equal to = l . l The coordinates of the point radians along the unit circle are , Since the first coordinate gives the cosine and the second the sine at x = cos = sin = . . , 140 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Similarly, a right triangle that contains 30° and 60° angles has sides that are in the ratio of 1: : 2 (side opposite 30° angle : side opposite 60° angle: hypotenuse). In order for such a triangle to fit into the unit circle, the hypotenuse must have length 1, so that the side opposite the 30° angle must be opposite the 60° angle must be and the side . If the triangle is fit into the unit circle so that the 30° angle is placed at the center of the circle, the distance around the circle from the initial point to the end of the hypotenuse is or The coordinates of the point giving cos = , and sin th of the circle, or radians. radians along the unit circle are = , , . On the other hand, if the triangle is fit into the unit circle so that the 60° angle is placed at the center of the circle, the distance around the circle from the initial point to the end of the hypotenuse is radians. or th of the circle, that is, 2 1 CHAPTER 8: TRIGONOMETRIC FUNCTIONS The coordinates of the point you have cos = , and sin radians along the unit circle are = , 141 and . To summarize, you should now be familiar with the cosines and sines of the following x-values: You should make an effort to remember the information in this table. The values of the sines and cosines for these x-values and their integer multiples are the only ones that you need to know for your trigonometric work. By using the unit circle, you can also determine the cosine and sine of any integer multiple of these x-values. For example, to find sin , you travel in a clockwise direction around the unit circle once (a distance of , or ), then more in the same clockwise direction. Since you end at the same position you reach traveling in the positive direction, sin = sin = –1. 142 CHAPTER 8: TRIGONOMETRIC FUNCTIONS To find cos , you move around the circle into the second quadrant. After traveling a distance of around the circle you find yourself at a point with a negative first coordinate. Since you have gone past x = , and the unit circle is a symmetric figure, you are now as far to the left of the center of the circle as you would be to the right of the center for x = . From this you can see that cos Find sin = –cos = . . You want the second coordinate at the point you reach after traveling a distance around the circle in the positive direction. This puts you in the third quadrant, where the sine is negative. You are now as far below 0 as the point of the way around is above 0. Therefore, sin Find cos = –sin = . . You must now travel in the counterclockwise direction a distance of additional . Go once completely around the circle ( ) and an past where you started. The first coordinate at this point is the same as the first coordinate of the point that is from the initial point in the positive direction. cos = cos = . You can find the value of the trigonometric functions for any integer multiples of , , , and Find: sin , sin in the same way. Move around the circle the distance you wish and see which quadrant you stop in. Let the circle show you which sign is the proper one to use for your value. Ans.: , , , cos CHAPTER 8: TRIGONOMETRIC FUNCTIONS Graphing Sine and Cosine Using the information from the first part of this chapter and the unit circle, you can obtain graphs for the sine and cosine functions. Both sine and cosine are defined for all x-values, since you can travel any distance you wish in either direction around the circle, and the point you reach has both a first and a second coordinate defined. To graph f1(x) = sin x, you may first find the intercept point. Earlier you saw in the unit circle that sin 0 = 0. You cannot say much about end-behavior, because the sine function is cyclic. Moving through larger x-values takes you around the circle again, either in the positive or negative direction. However, all values for the sine function are between –1 and 1. You have already seen that the sine function is an odd function, and that it is cyclic with a period of . Therefore, when you get the graph for xvalues between x = 0 and x = entire graph. , you know enough to determine the Remember that as your distance around the circle in the positive direction increases from 0 to , sin x increases from 0 to 1. At first this increase is rapid, because the circle rises steeply near your starting point. As you get closer to x = when x = , the sine increases at a slower rate until it reaches 1 . For distances around the circle greater than , the sine values decrease at the same rate that they previously increased, until the sine returns to 0 when x = . The sine values then continue to decrease until x = , where you reach the lowest sine value, –1. From this point, the sine values again increase, returning to 0 when x= . 143 144 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Once you have the graph from x = 0 to x = , you can either analyze the change in sine values as you move in the negative direction around the circle, or use the fact that the sine function is an odd function to obtain the graph for negative x-values from x = 0 to x = . Since the sine function is cyclic with a period of , you can obtain the graph for the entire domain by repeating what you already have for every additional distance of you move in either direction. Complete graph of f1(x) = sin x. Graphs having this regular, symmetric, cyclic pattern are called sinusoidal curves. Oscilloscopes often show such patterns, for example. Obtaining the graph of f2(x) = cos x requires a similar analysis. You can find the intercept by looking at the unit circle, cos 0 = 1. As you move away from your initial point around the circle in the positive direction, the first coordinate of your point decreases from 1, slowly at first, then more rapidly as you get near x = , where the cosine is 0. The cosine values continue to decrease rapidly, then more slowly as you get near x = , where the value of the cosine is –1. From x = to x = , the cosine values increase at the same rate they decrease between 0 and . At the end of one complete circuit around the unit circle the cosine is again equal to 1. This gives you the graph of one period of f2(x) = cos x. CHAPTER 8: TRIGONOMETRIC FUNCTIONS 145 To obtain the graph in the negative direction, you can use the fact that the cosine is an even function and produce the other side of the graph using the symmetry, or you can analyze the change in the cosine values as you move around the circle in the counterclockwise direction. The cosines decrease from 1 down to –1, which the function reaches when x = , then increase up to 1 again at x = . Using more negative x-values than this starts you into the next cycle. You obtain the complete graph by repeating this graph through its next cycles both for positive and negative x-values. Again, end-behavior is irrelevant because the cosine function is cyclic. Complete graph of f2(x) = cos x. Since sin (x + ) = cos x, you can also obtain the graph of the cosine function by shifting the graph of the sine function units to the left. Graph of cos x superimposed on the graph of sin x. This property was established on page 138. 146 CHAPTER 8: TRIGONOMETRIC FUNCTIONS You can also obtain the graphs of trigonometric functions modified in other ways besides linear translations. Sketch the graphs of a. –5 cos x b. sin x. Multiplying the sine or cosine function by a number changes the range of possible values [–1, 1] to multiples of these values. Such a multiple is called the amplitude of the function. For example, the graph of f3(x) = 4 cos x looks the same as the graph of cos x, but each value is 4 times greater, so that the maximum and minimum values for f3(x) are 4 and –4. (a) (b) Graph of 4 cos x compared to cos x. Multiplying the variable by a number before taking the sine or cosine changes the speed you travel around the circle, and therefore changes the period of each cycle of the function. Sketch the graphs of: a. cos x b. 3 sin 4x For example, let f4(x) = sin 2x. Since each x-value is doubled, you are twice as far along the circle as you would be in sin x for the same x-value. You return to your initial point on the circle in one-half the time. As x changes from 0 to , 2x changes from 0 to . The period of this function is rather than . Otherwise, the graph looks the same as sin x with maximum value 1 and minimum value –1. (a) Graph of sin 2x compared to sin x. (b) CHAPTER 8: TRIGONOMETRIC FUNCTIONS When you add or subtract a number from the trigonometric function, the entire graph rises or falls that number of units. Sketch the graph of + sin x. For example, let f5(x) = 2 sin x – 3. There are several steps to obtaining the graph of this function from the graph of sin x. The coefficient of x, , slows down your trip around the unit circle to half the usual speed. This makes the period twice as long as usual, since x must go from 0 to in order for x to travel from 0 to . The coefficient of sin x, 2, doubles the amplitude. The graph cycles between a low of –2 and a high of 2. Subtracting 3 from each trigonometric value moves the entire graph downward 3 units. Sketch the graphs of a. 2 cos (x – 4) b. sin (x + Graph of f5(x) compared to sin x. Another example that combines several of these elementary transformations is f6(x) = 1 – cos 3(x – ). Compared to the graph of cos x, f6(x) is reflected across the x-axis, has the amplitude reduced to , has the period changed to often, moves which makes the graph cycle 3 times as units to the right and 1 up. (a) (b) Graph of f6(x) compared to cos x. ) 147 148 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Other Trigonometric Functions The other trigonometric functions, tangent, cotangent, secant and cosecant, are defined in terms of the sine and cosine functions. You can obtain the properties and graphs of these functions by using your knowledge of the sine and cosine. Your variable still represents the distance traveled in the counterclockwise direction around a unit circle. The tangent (written tan x), cotangent (cot x), secant (sec x) and cosecant (csc x) are all defined as rational functions of the sine and cosine. tan x = , cot x = sec x = , csc x = Note that cot x = csc x = , not cos x = , not Placing a right triangle inside the unit circle, you can see that the tangent is the ratio of the two sides of the triangle that have lengths equal to sin x and cos x. These are the side opposite the angle placed at the center of the circle and the side adjacent to this angle. Similarly, the cotangent is the ratio of the length of the adjacent side to the length of the opposite side, the secant is 1 (the length of the hypotenuse of the triangle) divided by the length of the adjacent side, and the cosecant is the length of the hypotenuse divided by the length of the opposite side. These triangle relationships hold even when the triangles are not bounded by the unit circle. tan x = cot x = sec x = csc x = , but and . CHAPTER 8: TRIGONOMETRIC FUNCTIONS Graphs of the Other Trigonometric Functions You can obtain the graphs for tan x, cot x, sec x and csc x by using the properties you have already derived from the unit circle for sin x and cos x. For the graph of tan x, note that you have a rational function, with cos x in the denominator. The first thing you do to graph any rational function is to eliminate any values of the input variable which would produce a value of zero in the denominator. Looking at the unit circle, you can see that cos x is zero each time you reach the end of the first and third quadrant, traveling in the positive direction. The values of x at these points include , and all the other odd integer multiples of The intercept, tan 0 = = , , . Traveling in the clockwise direction, you reach the same points when x is other negative odd integer multiples of , , , and all the . = 0. Notice that tan x is also 0 when x = , , , and any other even integer multiple of because when you travel these distances around the unit circle you return to the initial point where the sine is 0 and the cosine is 1. You can also see that tan x =0 at the odd multiples of , since the sine is also equal to 0 there, while the cosine is –1. Since these are the only places where the sine is 0, they are the only values for which the tangent is equal to 0. The graph of the tangent crosses the axis when x is an integer multiple of and at no other values. When you use values just before x = , still in the first quadrant, both sine and cosine are positive and so is the tangent. However, the cosine is close to 0, so the tangent is a large positive number for these values. On the other side of x = , the cosine is negative although the sine is still positive (near 1), causing the tangent to be negative. The tangent is again large in size, since the cosine is close to 0 if you have not gone far past x = . Continuing around the circle, the tangent stays negative through the second quadrant, but becomes less negative (closer to 0) as the sine approaches 0 and the cosine nears –1. As you have already seen, the tangent equals 0 at x = . Continuing into the third quadrant, the tangent becomes positive, since both sine and cosine are negative. 149 150 CHAPTER 8: TRIGONOMETRIC FUNCTIONS The tangent values are increasing since the sine heads toward –1 while the cosine becomes closer to 0. Dividing by the small cosine values gives you large positive quotients for the tangent values as the x-values get closer to x= , which is a value that is not in the domain of tan x. On the other side of x = , the tangent is negative, since the sine is negative and the cosine positive. Near x = , the tangent has very negative values, since the cosine in the denominator is close to 0, but as the cosine increases toward 1 and the sine approaches 0, the tangent also approaches a value of 0, which it attains when x = . Now you are back to where you started, and the values for the tangent cycle through the same sequence of values as x increases from to . You started a new cycle of values at x = , and you can see from the graph you have drawn that the period of the tangent function is rather than the period of both sine and cosine. Going in the negative direction, you obtain the same tangent values, but in the opposite order. The tangent is negative in the fourth quadrant, positive in the third, then negative in the second and positive in the first. When you return to the initial point after traveling a distance of , you have completed two cycles and are back where you started. Complete graph of tan x. Notice that the tangent can take on any positive or negative value, unlike sine and cosine, which must have values between –1 and 1. CHAPTER 8: TRIGONOMETRIC FUNCTIONS Since sin (–x) = –sin x and cos (–x) = cos x, tan (–x) = = = –tan x. The tangent function is an odd function, as you can also see from the graph. Cot x = is also an odd function, and has a graph that you can obtain from that of the tangent by noting that both tan x and cot x have the same signs in each quadrant, but the cotangent is large when the tangent is near zero, and vice-versa. Analyzing cot x = separately from tan x, you first note that any value of x for which sin x = 0 is not in the domain of cot x. The points at which sin x is zero include x = 0, , , , and all other integer multiples of . There is no intercept. The cotangent function has a value of zero when cos x is zero. This occurs when x is any odd integer multiple of . Note that this is exactly the opposite of the situation for the tangent function, which is the reciprocal of the cotangent. As you move through the first quadrant, cos x decreases from 1 to 0 and sin x increases from 0 to 1. Since the numerator of cot x is decreasing while the denominator is increasing, the function values rapidly decrease to 0, which you reach when x = . In the second quadrant, the cotangent becomes negative, and since the sin x term in the denominator decreases, the size of these negative values becomes greater as you approach x = , for which the function does not exist. In the third quadrant, both sine and cosine are negative. The cosine heads toward zero while the size of the sine increases toward –1. This combination produces positive cotangent values that decrease to 0 at x = . When you pass into the fourth quadrant, the cotangent values become increasingly negative again since the sine remains negative, but gets closer to zero. This makes the cotangent values drop off rapidly in the negative direction. As at x= 0, cot x does not exist at x= , but beyond this value it cycles through the same values as on your preceding trip around the circle. Moving in the clockwise direction produces the same results in the opposite order. 151 152 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Complete graph of cot x. Note from the graph that, like tan x, the period of cot x is that cot x is another odd function. To obtain the graph of sec x, use its definition sec x = . You can also see . You must eliminate the same points from the domain of sec x as from tan x, since both these functions have a denominator of cos x. This means that again you have vertical asymptotes at all the odd multiples of The intercept is = . = 1. You also obtain this value whenever x is an integer multiple of . Since all cosine values lie between –1 and 1, the secant values are reciprocals of those numbers, namely numbers greater than or equal to 1 and less than or equal to –1. Since cosine values are never greater than 1 nor less than –1, secant values are never between –1 and 1. As you move around the first quadrant, the cosine values decrease from 1 to 0. Simultaneously, their reciprocal values increase from 1. Secant values are steadily increasing from 1 until you approach x = , for which there is no secant value defined. In the second quadrant, cosine (and therefore secant) values are negative. The cosine values are near zero at first, causing the secant to be extremely negative, then the cosine approaches –1, which it reaches at x = . The secant meanwhile becomes less negative until it also reaches –1 when you get to x = . cot (–x) = = = –cot x CHAPTER 8: TRIGONOMETRIC FUNCTIONS 153 The cosine remains negative in the third quadrant. It retraces the same values (in the opposite order) that has in the second quadrant. So does the secant, although the secant does not exist at x = and becomes extremely negative as you approach this x-value. In the fourth quadrant, the cosine returns to positive values, increasing to 1 at x= . The secant is also positive, very large when x is just past , and decreasing to reach 1 when x = . The secant also repeats its values again as you make the next circuit of the circle. Proceeding in the counterclockwise direction again produces all the same values in the opposite order. Complete graph of sec x. Another way to obtain the graph of the secant is to take advantage of the fact that sec x is the reciprocal of cos x. Any time that a function reaches the value of zero, its reciprocal does not exist. At a point eliminated from the domain of the function because it causes a division by zero, the reciprocal value is zero). When a function is equal to either of the values –1 or 1, so is the reciprocal. The reciprocal function is positive or negative when the original function is. Input values that produce large output values for the original function are small (closer to zero) in the reciprocal and vice-versa. Using these facts allows you to sketch the graph of a reciprocal function with the graph of the original by intersecting them at function values of –1 and 1, producing vertical asymptotes for one when the other is zero and large values when the other is small, keeping their signs the same. One example of such a graph is on the right and the graph of sec x drawn as the reciprocal of cos x is shown on the following page. Graph of a function and its reciprocal. 154 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Sketch the graph of csc x. Graph of csc x given with sin x. Graph of sec x drawn as the reciprocal of cos x. The graphs of tangents, cotangents, secants and cosecants change by translations and stretches in the same ways that the sine and cosine graphs do. The graph of tan x looks like that of tan x, but the coefficient of x, , decreases your speed moving around the unit circle. Instead of a period of , you have a period of . The graph does not complete a full cycle until the x-values have gone through an interval of width Graph of tan .25x. . Graph of tan x drawn with the same scale.