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4 Trigonometry and Complex Numbers

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4 Trigonometry and Complex Numbers 4 Trigonometry and Complex Numbers Trigonometry developed from the study of triangles, particularly right triangles, and the relations between the lengths of their sides and the sizes of their angles. The trigono- metric functions that measure the relationships between the sides of similar triangles have far-reaching applications that extend far beyond their use in the study of triangles. Complex numbers were developed, in part, because they complete, in a useful and ele- gant fashion, the study of the solutions of polynomial equations. Complex numbers are useful not only in mathematics, but in the other sciences as well. Trigonometry Most of the trigonometric computations in this chapter use six basic trigonometric func- tions The two fundamental trigonometric functions, sine and cosine, can be de¿ned in terms of the unit circle—the set of points in the Euclidean plane of distance one from the origin. A point on this circle has coordinates +frv w> vlq w,,wherew is a measure (in radians) of the angle at the origin between the positive {-axis and the ray from the ori- gin through the point measured in the counterclockwise direction. The other four basic trigonometric functions can be de¿ned in terms of these two—namely, vlq { 4 wdq { @ vhf { @ frv { frv { frv { 4 frw { @ fvf { @ vlq { vlq { 3 ?w? For 5 , these functions can be found as a ratio of certain sides of a right triangle that has one angle of radian measure w. Trigonometric Functions The symbols used for the six basic trigonometric functions—vlq, frv, wdq, frw, vhf, fvf—are abbreviations for the words cosine, sine, tangent, cotangent, secant, and cose- cant, respectively.You can enter these trigonometric functions and many other functions either from the keyboard in mathematics mode or from the dialog box that drops down when you click or choose Insert + Math Name. When you enter one of these functions from the keyboard in mathematics mode, the function name automatrically turns gray when you type the ¿nal letter of the name. 90 Chapter 4 Trigonometry and Complex Numbers Note Ordinary functions require parentheses around the function argument, while trigonometric functions commonly do not. The default behavior of your system allows trigonometric functions without parentheses. If you want parentheses to be required for all functions, you can change this behavior in the Maple Settings dialog. Click the Def- inition Options tab and under Function Argument Selection Method, check Convert Trigtype to Ordinary. For further information see page 126. To ¿nd values of the trigonometric functions, use Evaluate or Evaluate Numeri- cally. L Evaluate s s 6 4 4 vlq 7 @ 5 5 vlq +4, @ vlq 4 vlq 93 @ 5 6 L Evaluate Numerically 6 vlq 7 @ = :3:44 vlq +4, @ = ;747: vlq 93 @ = ;9936 The notation you use determines whether the argument is interpreted as radians or degrees: vlq 63 @ =<;;36 and vlq 63 @ =8. The degree symbol can be either a small green or red circle. The small green circle is entered from the Insert + Unit Name dialog. The small red circle appears on the Common Symbols toolbar and on the Binary Operations symbol panel, and must be entered as a superscript. With no symbol, the argument is interpreted as radians, and with either a green or red degree symbol, the argument is interpreted as degrees. All operations will convert angle measure to radians. See page 91 for a discussion of conversion. See page 37 for a discussion of units that can be used for plane angles. Your choice for Digits Used in Display in the Maple + Settings dialog determines the number of places displayed in the response to Evaluate Numerically. Expression Evaluates Evaluate Numerically 4 vlq 7 5 5 =:3:44 ;:: vlq 476:3 vlq =58568 43;33 oq +vlq {, orj vlq { =7675< oq +vlq {, 43 oq5.oq8 46 6873 9=;39; 43Ã5 933 Solving Trigonometric Equations You can use both Exact and Numeric from the Solve submenu to ¿nd solutions to trigonometric equations. These operations also convert degrees to radians. Use of deci- mal notation in the equation gives you a numerical solution, even with Exact. Trigonometry 91 Equation Solve +s Exact Solve + Numeric 4 { @ vlq 7 { @ 5 5 { @ =:3:44 47 47 vlq 55 @ f @ f @6:=6:6 f 44 vlq <3 Ã5 orj43 vlq { @ 4=49:< { @9=:<;; 43 { @6=3:69 3 46 Ã5 { @6 87 { @ 933 {@9=;39; 43 Note that the answers are different for the equation orj43 vlq { @ 4=49:<.Thisdif- ference occurs because there are multiple solutions and the two commands are ¿nding different solutions. The Numeric command from the Solve submenu offers the ad- vantage that you can specify a range in which you wish the solution to lie. Enter the equation and the range in different rows of a display or a one-column matrix. L Solve + Numeric { @ 43 vlq { i{ @:=39;5j { 5 +8> 4, ,Solutionis: The interval +8> 4, was speci¿ed for the solution in the preceding example. By specifying other intervals, you can ¿nd all seven solutions: i{ @3j, i{ @ 5=;856j, i{ @ :=39;5j, i{ @ ;=7565j, as depicted in the following graph. The Exact com- mand for solving equations gives only the solution { @3for this equation. 10 5 -10 -50 5 10 x -5 -10 When any operation is applied to an angle represented in degrees by a mathematics superscript, such as 7;, degrees are automatically converted to radians. To go in the other direction, replace 5 radians with 693 and convert other angles proportionately. You can also solve directly for the number of degrees. Both methods follow. L To convert radians to degrees (using ratios) { @ { 1. Write the equation 693 5 ,where represents radians. 2. Leave the insertion point in this equation. 3. From the Solve submenu, choose Exact or Numeric. 92 Chapter 4 Trigonometry and Complex Numbers 4. Name as the Variable to Solve for. 4;3 @ { 5. Choose OK to get . L To convert radians to degrees (directly) 1. Write an equation such as 5@ ,or(usingInsert + Unit Name) 5udg@ . 2. Leave the insertion point in this equation. 693 3. From the Solve submenu, choose Exact or Numeric,toget @ or @447= 8< 46 { @ Example 17 Convert 933 radians to degrees as follows. 46 @ 933 1. Write the equation 693 5 . 2. Leave the insertion point in this equation. 6< 3. From the Solve submenu, choose Exact to get @ 43 degrees, or choose Numeric to get @6=< degrees= -or- 46 @ 1. Write the equation 933 2. Leave the insertion point in this equation 6< 3. From the Solve submenu, choose Exact to get @ 43 , or choose Numeric to get @6=<. See page 43 for additional examples of converting units. L To change 3=< degrees to minutes Apply Evaluate to 3=< 93 to get 3=< 93 @ 87=3 -or- Enter degree and minute symbols from the Unit Name dialog. Apply Solve to 3=< @ { 3 to get { @87=3 Trigonometry 93 To check these results, apply Evaluate to 6873 or 6 87 3 to get 46 6873 @ 6 87 3 @9= ;39 ; 43Ã5 udg 933 or Trigonometric Identities This section illustrates the effects of some operations on trigonometric functions. First, simpli¿cations and expansions of various trigonometric expressions illustrate many of the familiar trigonometric identities. De¿nitions in Terms of Basic Trigonometric Functions Apply Simplify to the secant, and cosecant to ¿nd their de¿nition in terms of the sine and cosine functions. L Simplify 4 4 vhf { @ fvf { @ frv { vlq { Pythagorean Identities L Simplify vlq5 { .frv5 { @ 4 wdq5 { vhf5 { @ 4 frw5 { fvf5 { @ 4 Addition Formulas L Expand vlq +{ . |, @ vlq { frv | .frv{ vlq | frv +{ . |, @ frv { frv | vlq { vlq | wdq { .wdq| wdq +{ . |,@ 4 wdq { wdq | Apply Combine + Trig Functions to the expansions of vlq +{ . |, and frv +{ . |, to return them to their original form and to change the expansion of wdq +{ . |, to the vlq+{.|, form frv+{.|, . 94 Chapter 4 Trigonometry and Complex Numbers Double-Angle Formulas L Expand vlq 5 @ 5 vlq frv frv 5 @ 5 frv5 4 frv wdq 5 @ 5 +vlq , 5frv5 4 You can uncover other multiple-angle formulas with Expand. Following are some examples. L Expand vlq 9 @65vlq frv8 65 vlq frv6 .9vlq frv vlq 57 @ ;6;;93; vlq frv56 7946:677 vlq frv54 . 4434337;3 vlq frv4< 47<7553;3 vlq frv4: . 45:33;:9; vlq frv48 :34;<389 vlq frv46 . 5867937; vlq frv44 8;8:5;3 vlq frv< . ;569;3 vlq frv: 97397 vlq frv8 .55;;vlq frv6 57 vlq frv vlq+5d .6e, @ ; vlq d frv d frv6 e 9vlqd frv d frv e . ; frv5 d vlq e frv5 e 5 frv5 d vlq e 7 vlq e frv5 e . vlq e Combining and Simplifying Trigonometric Expressions Products and powers of trigonometric functions and hyperbolic functions are combined into a sum of trigonometric functions or hyperbolic functions whose arguments are in- tegral linear combinations of the original arguments. L Combine + Trig Functions 4 4 5 4 4 vlq { vlq | @ 5 frv +{ |, 5 frv +{ . |,vlq{ @ 5 5 frv 5{ 4 4 5 4 4 vlq { frv | @ 5 vlq +{ . |,. 5 vlq +{ |,frv{ @ 5 frv 5{ . 5 8 8 4 8 8 vlq { frv { @ 845 vlq 43{ . 589 vlq 5{ 845 vlq 9{ 6 6 4 6 4 vlqk { frvk { @ 65 vlqk 9{ 65 vlqk 5{ vlqk { frvk { @ 5 vlqk 5{ The command Simplify combines and simpli¿es trigonometric expressions, as in the Trigonometry 95 following examples. L Simplify 5 4 5 5 5 5 5 frv { . 7 vlq 5{ vlq { frv { . 5 vlq { @ frv { .5 +frv 6w . 6 frv w, vhf w @ 7 frv5 w vlq 6d . 7 vlq6 d @ 6 vlq d wdq 5 vlq 5 .
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