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14.33 2 tan x sin x 14.34 2 cos x 14.35 2 sin x 14.36 sin x 14.37 sec x 14.38 2 cos x 14.39 3 sin x 1 1 sec x csc x
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CHAPTER 14 Trigonometric Equations
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14.40 14.41 14.42 14.43 14.44 14.45 14.46 14.47 1 sin x 2 cos x. 4 cos x 23/2. 1. 23/2. 1/ 23. 1/2. 1. 0 and 0 2. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. 0.64, 3 /2 1.80, 5.76 2 /3, 5 /6, 5 /3, 11 /6 /12, 5 /12, 3 /4, 13 /12, 17 /12, 7 /4 /3 No solution in given interval /4, 5 /4 0, /3, 5 /3
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Solve each of the following systems for r 14.48 r r r r a sin a cos 2 a cos a sin 2
<2 . Ans. /6, r /6, r /2, r /6, r /6, r a/2 a/2; 0;
3 ,r 3 ,r 0
Ans.
23a/2 23a/2 6 6
4(1 cos ) 3 sec
Ans.
/3, r /3, r
Solve each of the following equations. 14.51 14.52 14.53 Arctan 2x Arcsin x Arccos x Arctan x Arctan x Arctan x /4. /2. /2. Ans. Ans. Ans. x x x 0.2808 0.7862 0
Complex Numbers
15.1 Imaginary Numbers
The square root of a negative number (e.g., 2 1, 2 5, and 2 9 ) is called an imaginary number. Since by definition 2 5 25 # 2 1 and 2 9 29 # 2 1 3 2 1, it is convenient to introduce the symbol i 2 1 and to adopt 2 5 i 25 and 2 9 3i as the standard form for these numbers. The symbol i has the property i2 1; and for higher integral powers we have i3 i2 i ( 1)i i, 4 2 2 2 5 4 i (i ) ( 1) 1, i i i i, etc. The use of the standard form simplifies the operations on imaginary numbers and eliminates the possibility of certain common errors. Thus 2 9 # 24 6i since 2 9 # 24 3i(2) 6i but 2 36 2 9 # 2 4 2 236 since 2 9 # 2 4 (3i)(2i) 6i2 6.
15.2 Complex Numbers
A number a bi, where a and b are real numbers, is called a complex number. The first term a is called the real part of the complex number, and the second term bi is called the imaginary part. Complex numbers may be thought of as including all real numbers and all imaginary numbers. For example, 5 5 0i and 3i 0 3i. Two complex numbers a bi and c di are said to be equal if and only if a c and b d. The conjugate of a complex number a bi is the complex number a bi. Thus, 2 3i and 2 3i, and 3 4i and 3 4i are pairs of conjugate complex numbers.
15.3 Algebraic Operations
Addition
To add two complex numbers, add the real parts and the imaginary parts separately.
EXAMPLE 15.1 (2
Subtraction
To subtract two complex numbers, subtract the real parts and the imaginary parts separately.
EXAMPLE 15.2 (2
( 5)]i
Multiplication
To multiply two complex numbers, carry out the multiplication as if the numbers were ordinary binomials and replace i2 by 1.
EXAMPLE 15.3 (2
3i)(4
15i2
15( 1)
CHAPTER 15 Complex Numbers
Division
To divide two complex numbers, multiply both numerator and denominator of the fraction by the conjugate of the denominator.
EXAMPLE 15.4
3i 5i
(2 (4
3i)(4 5i)(4 7 41
5i) 5i) 22i nor
15) 16
(10 25
12i)
7 41
22 i. 41
[Note the form of the result; it is neither
1 ( 7 41
22i).]
(See Probs. 15.1 to 15.9.)
15.4 Graphic Representation of Complex Numbers
The complex number x yi may be represented graphically by the point P [(see Fig. 15.1(a)] whose rectangular coordinates are (x, y). The point O having coordinates (0, 0) represents the complex number 0 0i 0. All points on the x axis have coordinates of the form (x, 0) and correspond to real numbers x 0i x. For this reason, the x axis is called the axis of reals. All points on the y axis have coordinates of the form (0, y) and correspond to imaginary numbers 0 yi yi. The y axis is called the axis of imaginaries. The plane on which the complex numbers are represented is called the complex plane. In addition to representing a complex number by a point P in the complex plane, the number may be represented [see Fig. 15.1(b)] by the directed line segment or vector OP.
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