Cos 45 sin 45: Mathway | Популярные задачи

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Mathway | Популярные задачи

1Найти точное значениеsin(30)
2Найти точное значениеsin(45)
3Найти точное значениеsin(30 град. )
4Найти точное значениеsin(60 град. )
5Найти точное значениеtan(30 град. )
6Найти точное значениеarcsin(-1)
7Найти точное значениеsin(pi/6)
8
Найти точное значение		cos(pi/4)
9Найти точное значениеsin(45 град. )
10Найти точное значениеsin(pi/3)
11Найти точное значениеarctan(-1)
12Найти точное значениеcos(45 град. )
13Найти точное значениеcos(30 град. )
14Найти точное значениеtan(60)
15
Найти точное значение		csc(45 град. )
16Найти точное значениеtan(60 град. )
17Найти точное значениеsec(30 град. )
18Найти точное значениеcos(60 град. )
19Найти точное значениеcos(150)
20Найти точное значениеsin(60)
21Найти точное значениеcos(pi/2)
22Найти точное значениеtan(45 град. )
23Найти точное значениеarctan(- квадратный корень из 3)
24Найти точное значениеcsc(60 град. )
25Найти точное значениеsec(45 град. )
26Найти точное значениеcsc(30 град. )
27Найти точное значение sin(0)
28Найти точное значениеsin(120)
29Найти точное значениеcos(90)
30Преобразовать из радианов в градусыpi/3
31Найти точное значениеtan(30)
32Преобразовать из градусов в радианы45
33Найти точное значениеcos(45)
34Упроститьsin(theta)^2+cos(theta)^2
35Преобразовать из радианов в градусыpi/6
36Найти точное значениеcot(30 град. )
37Найти точное значениеarccos(-1)
38Найти точное значениеarctan(0)
39Найти точное значениеcot(60 град. )
40Преобразовать из градусов в радианы30
41Преобразовать из радианов в градусы(2pi)/3
42Найти точное значениеsin((5pi)/3)
43Найти точное значениеsin((3pi)/4)
44Найти точное значениеtan(pi/2)
45Найти точное значениеsin(300)
46Найти точное значениеcos(30)
47Найти точное значениеcos(60)
48Найти точное значениеcos(0)
49Найти точное значениеcos(135)
50Найти точное значениеcos((5pi)/3)
51Найти точное значениеcos(210)
52Найти точное значениеsec(60 град. )
53Найти точное значениеsin(300 град. )
54Преобразовать из градусов в радианы135
55Преобразовать из градусов в радианы150
56Преобразовать из радианов в градусы(5pi)/6
57Преобразовать из радианов в градусы(5pi)/3
58Преобразовать из градусов в радианы89 град.
59Преобразовать из градусов в радианы60
60Найти точное значениеsin(135 град. )
61Найти точное значениеsin(150)
62Найти точное значениеsin(240 град. )
63Найти точное значениеcot(45 град. )
64Преобразовать из радианов в градусы(5pi)/4
65Найти точное значениеsin(225)
66Найти точное значениеsin(240)
67Найти точное значениеcos(150 град. )
68Найти точное значениеtan(45)
69Вычислитьsin(30 град. )
70Найти точное значениеsec(0)
71Найти точное значениеcos((5pi)/6)
72Найти точное значениеcsc(30)
73Найти точное значениеarcsin(( квадратный корень из 2)/2)
74Найти точное значениеtan((5pi)/3)
75Найти точное значениеtan(0)
76Вычислитьsin(60 град. )
77Найти точное значениеarctan(-( квадратный корень из 3)/3)
78Преобразовать из радианов в градусы(3pi)/4
79Найти точное значениеsin((7pi)/4)
80Найти точное значениеarcsin(-1/2)
81Найти точное значениеsin((4pi)/3)
82Найти точное значениеcsc(45)
83Упроститьarctan( квадратный корень из 3)
84Найти точное значениеsin(135)
85Найти точное значениеsin(105)
86Найти точное значениеsin(150 град. )
87Найти точное значениеsin((2pi)/3)
88Найти точное значение tan((2pi)/3)
89Преобразовать из радианов в градусыpi/4
90Найти точное значениеsin(pi/2)
91Найти точное значениеsec(45)
92Найти точное значениеcos((5pi)/4)
93Найти точное значениеcos((7pi)/6)
94Найти точное значениеarcsin(0)
95Найти точное значение
sin(120 град. )
96Найти точное значениеtan((7pi)/6)
97Найти точное значениеcos(270)
98Найти точное значениеsin((7pi)/6)
99Найти точное значениеarcsin(-( квадратный корень из 2)/2)
100Преобразовать из градусов в радианы88 град.

trigonometry full — презентация онлайн

7.1
Trigonometric Identities

2. Trigonometric Identities

We know that an equation is a statement that two
mathematical expressions are equal. For example, the
following are equations:
x+2=5
(x + 1)2 = x2 + 2x + 1
sin2 t + cos2 t = 1.
An identity is an equation that is true for all values of the
variable(s). The last two equations above are identities, but
the first one is not, since it is not true for values of x other
than 3.
2

3. Trigonometric Identities

A trigonometric identity is an identity involving
trigonometric functions. We begin by listing some of the
basic trigonometric identities.
3

4. Simplifying Trigonometric Expressions

Identities enable us to write the same expression in
different ways. It is often possible to rewrite a
complicated-looking expression as a much simpler one.
To simplify algebraic expressions, we used factoring,
common denominators, and the Special Product Formulas.
To simplify trigonometric expressions, we use these same
techniques together with the fundamental trigonometric
identities.
4

5. Example 1 – Simplifying a Trigonometric Expression

Simplify the expression cos t + tan t sin t.
Solution:
We start by rewriting the expression in terms of sine and
cosine.
cos t + tan t sin t = cos t +
sin t
Reciprocal identity
=
Common denominator
=
Pythagorean identity
= sec t
Reciprocal identity
5
Proving Trigonometric Identities
6

7. Proving Trigonometric Identities

Many identities follow from the fundamental identities.
In the examples that follow, we learn how to prove that a
given trigonometric equation is an identity, and in the
process we will see how to discover new identities.
First, it’s easy to decide when a given equation is not an
identity.
All we need to do is show that the equation does not hold
for some value of the variable (or variables).
7

8. Proving Trigonometric Identities

Thus the equation
sin x + cos x = 1
is not an identity, because when x = /4, we have
To verify that a trigonometric equation is an identity, we
transform one side of the equation into the other side by a
series of steps, each of which is itself an identity.
8

9. Proving Trigonometric Identities

9

10. Example 2 – Proving an Identity by Rewriting in Terms of Sine and Cosine

Consider the equation cos (sec – cos ) = sin2 .
(a) Verify algebraically that the equation is an identity.
(b) Confirm graphically that the equation is an identity.
Solution:
(a) The left-hand side looks more complicated, so we start
with it and try to transform it into the right-hand side:
LHS = cos (sec – cos )
= cos
Reciprocal identity
10

11. Example 2 – Solution

cont’d
= 1 – cos2
Expand
= sin2 = RHS
Pythagorean identity
(b) We graph each side of the equation to see whether the
graphs coincide. From Figure 1 we see that the graphs
of y = cos (sec – cos ) and y = sin2 are identical.
This confirms that the equation
is an identity.
Figure 1
11

12. Proving Trigonometric Identities

In Example 2 it isn’t easy to see how to change the
right-hand side into the left-hand side, but it’s definitely
possible. Simply notice that each step is reversible.
In other words, if we start with the last expression in the
proof and work backward through the steps, the right-hand
side is transformed into the left-hand side.
You will probably agree, however, that it’s more difficult to
prove the identity this way. That’s why it’s often better to
change the more complicated side of the identity into the
simpler side.
12

13. Proving Trigonometric Identities

In Example 3 we introduce “something extra” to the
problem by multiplying the numerator and the denominator
by a trigonometric expression, chosen so that we can
simplify the result.
13

14. Example 3 – Proving an Identity by Introducing Something Extra

Verify the identity
= sec u + tan u.
Solution:
We start with the left-hand side and multiply the numerator
and denominator by 1 + sin u:
LHS =
=
Multiply numerator and
denominator by 1 + sin u
14

15.

Example 3 – Solutioncont’d
=
Expand denominator
=
Pythagorean identity
=
Cancel common factor
=
Separate into two fractions
= sec u + tan u
Reciprocal identities
15

16. Proving Trigonometric Identities

Here is another method for proving that an equation is an
identity.
If we can transform each side of the equation separately,
by way of identities, to arrive at the same result, then the
equation is an identity. Example 6 illustrates this procedure.
16

17. Example 4 – Proving an Identity by Working with Both Sides Separately

Verify the identity
Solution:
We prove the identity by changing each side separately
into the same expression. (You should supply the reasons
for each step.)
LHS =
= sec + 1
17

18. Example 4 – Solution

RHS =
cont’d
=
= sec + 1
It follows that LHS = RHS, so the equation is an identity.
18

19. Proving Trigonometric Identities

We conclude this section by describing the technique of
trigonometric substitution, which we use to convert
algebraic expressions to trigonometric ones. This is often
useful in calculus, for instance, in finding the area of a
circle or an ellipse.
19

20. Example 5 – Trigonometric Substitution

Substitute sin for x in the expression
simplify. Assume that 0 /2.
, and
Solution:
Setting x = sin , we have
Substitute x = sin
Pythagorean identity
cos
Take square root
The last equality is true because cos 0 for the values of
in question.
20
7.2
Addition and
Subtraction Formulas
21

22. Addition and Subtraction Formulas

We now derive identities for trigonometric functions of
sums and differences.
22

23. Example 1 – Using the Addition and Subtraction Formulas

Find the exact value of each expression.
(a) cos 75
(b) cos
Solution:
(a) Notice that 75 = 45 + 30 . Since we know the exact
values of sine and cosine at 45 and 30 , we use the
Addition Formula for Cosine to get
cos 75 = cos (45 + 30 )
= cos 45 cos 30 – sin 45 sin 30
=
23

24.

Example 1 – Solution(b) Since
gives
cont’d
the Subtraction Formula for Cosine
cos
= cos
= cos
cos
+ sin
sin
24

25. Example 2 – Proving a Cofunction Identity

Prove the cofunction identity cos
= sin u.
Solution:
By the Subtraction Formula for Cosine we have
cos
= cos
cos u + sin
sin u
= 0 cos u + 1 sin u
= sin u
25

26. Addition and Subtraction Formulas

The cofunction identity in Example 3, as well as the other
cofunction identities, can also be derived from the following
figure.
cos
=
= sin u
The next example is a typical use of the Addition and
Subtraction Formulas in calculus.
26

27. Example 3 – An identity from Calculus

If f(x) = sin x, show that
Solution:
Definition of f
Addition Formula
for Sine
27

28. Example 3 – Solution

cont’d
Factor
Separate the fraction
28
Evaluating Expressions Involving
Inverse Trigonometric Functions
29

30.

Evaluating Expressions Involving Inverse Trigonometric FunctionsExpressions involving trigonometric functions and their
inverses arise in calculus. In the next examples we
illustrate how to evaluate such expressions.
30

31. Example 4 – Simplifying an Expression Involving Inverse Trigonometric Functions

Write sin(cos–1 x + tan–1 y) as an algebraic expression in x
and y, where –1 x 1 and y is any real number.
Solution:
Let = cos–1x and = tan–1y. We sketch triangles with
angles and such that cos = x and tan = y (see
Figure 2).
cos = x
tan = y
Figure 2
31

32. Example 4 – Solution

cont’d
From the triangles we have
sin =
cos =
sin =
From the Addition Formula for Sine we have
sin(cos–1 x + tan–1 y) = sin( + )
= sin cos + cos sin
Addition Formula for Sine
32

33. Example 4 – Solution

cont’d
From triangles
Factor
33
Expressions of the Form
A sin x + B cos x
34

35.

Expressions of the Form A sin x + B cos xWe can write expressions of the form A sin x + B cos x in
terms of a single trigonometric function using the Addition
Formula for Sine. For example, consider the expression
sin x +
cos x
If we set = /3, then cos =
can write
sin x +
and sin =
/2, and we
cos x = cos sin x + sin cos x
= sin(x + ) = sin
35

36. Expressions of the Form A sin x + B cos x

We are able to do this because the coefficients and
/2
are precisely the cosine and sine of a particular number, in
this case, /3.
We can use this same idea in general to write
A sin x + B cos x in the form k sin(x + ).
We start by multiplying the numerator and denominator
by
to get
A sin x + B cos x
=
36

37. Expressions of the Form A sin x + B cos x

We need a number with the property that
cos =
and
sin =
Figure 4 shows that the point (A, B) in the plane determines
a number with precisely this property.
Figure 4
37

38. Expressions of the Form A sin x + B cos x

With this we have
A sin x + B cos x =
=
(cos sin x + sin cos x)
sin(x + )
We have proved the following theorem.
38

39. Example 5 – A Sum of Sine and Cosine Terms

Express 3 sin x + 4 cos x in the form k sin(x + ).
Solution:
By the preceding theorem, k =
=
= 5.
The angle has the property that sin =
=
and cos =
= , and in Quadrant I (because sin
and cos are both positive), so = sin –1 . Using a
calculator, we get 53.1 .
Thus
3 sin x + 4 cos x 5 sin (x + 53.1 )
39

40. Example 5 – Graphing a Trigonometric Function

Write the function f(x) = –sin 2x +
cos 2x in the form
k sin(2x + ), and use the new form to graph the function.
Solution:
Since A = –1 and B =
, we have
k=
=
= 2.
The angle satisfies cos = – and sin =
/2. From
the signs of these quantities we conclude that is in
Quadrant II.
40

41. Example 5 – Solution

cont’d
Thus = 2 /3.
By the preceding theorem we can write
f(x) = –sin 2x +
cos 2x
= 2 sin
Using the form
f(x) = 2 sin 2
41

42. Example 5 – Solution

cont’d
We see that the graph is a sine curve with amplitude 2,
period 2 /2 = , and phase shift – /3. The graph is shown
in Figure 5.
Figure 5
42
7.3
Double-Angle, Half-Angle, and
Product-Sum Formulas
43

44. Double-Angle, Half-Angle, and Product-Sum Formulas

The identities we consider in this section are consequences
of the addition formulas. The Double-Angle Formulas
allow us to find the values of the trigonometric functions at
2x from their values at x.
The Half-Angle Formulas relate the values of the
trigonometric functions at x to their values at x. The
Product-Sum Formulas relate products of sines and
cosines to sums of sines and cosines.
44

45. Double-Angle Formulas

The formulas in the following box are immediate
consequences of the addition formulas.
45

46. Example 1– A Triple-Angle Formula

Write cos 3x in terms of cos x.
Solution:
cos 3x = cos(2x + x)
= cos 2x cos x – sin 2x sin x
Addition formula
= (2 cos2 x – 1) cos x
– (2 sin x cos x) sin x Double-Angle Formulas
= 2 cos3 x – cos x – 2 sin2 x cos x
Expand
46

47. Example 1 – Solution

cont’d
= 2 cos3 x – cos x – 2 cos x (1 – cos2 x) Pythagorean
identity
= 2 cos3 x – cos x – 2 cos x + 2 cos3 x
Expand
= 4 cos3 x – 3 cos x
Simplify
47

48. Double-Angle Formulas

Example 2 shows that cos 3x can be written as a
polynomial of degree 3 in cos x.
The identity cos 2x = 2 cos2 x – 1 shows that cos 2x is a
polynomial of degree 2 in cos x.
In fact, for any natural number n we can write cos nx as a
polynomial in cos x of degree n.
48
Half-Angle Formulas
49

50. Half-Angle Formulas

The following formulas allow us to write any trigonometric
expression involving even powers of sine and cosine in
terms of the first power of cosine only.
This technique is important in calculus. The Half-Angle
Formulas are immediate consequences of these formulas.
50

51. Example 2 – Lowering Powers in a Trigonometric Expression

Express sin2 x cos2 x in terms of the first power of cosine.
Solution:
We use the formulas for lowering powers repeatedly.
51

52. Example 2 – Solution

cont’d
Another way to obtain this identity is to use the
Double-Angle Formula for Sine in the form
sin x cos x = sin 2x. Thus
52

53. Half-Angle Formulas

53

54. Example 3 – Using a Half-Angle Formula

Find the exact value of sin 22.5 .
Solution:
Since 22.5 is half of 45 , we use the Half-Angle Formula
for Sine with u = 45 . We choose the + sign because 22.5
is in the first quadrant:
Half-Angle Formula
cos 45 =
54

55. Example 3 – Solution

cont’d
Common denominator
Simplify
55
Evaluating Expressions Involving
Inverse Trigonometric Functions
56

57.

Evaluating Expressions Involving Inverse Trigonometric FunctionsExpressions involving trigonometric functions and their
inverses arise in calculus. In the next example we illustrate
how to evaluate such expressions.
57

58. Example 4 – Evaluating an Expression Involving Inverse Trigonometric Functions

Evaluate sin 2 , where cos =
with in Quadrant II.
Solution :
We first sketch the angle in standard position with
terminal side in Quadrant II as in Figure 2.
Since cos = x/r = ,
we can label a side and the
hypotenuse of the triangle in
Figure 2.
To find the remaining side, we
use the Pythagorean Theorem.
Figure 2
58

59. Example 4 – Solution

x2 + y2 = r2
cont’d
Pythagorean Theorem
(–2)2 + y2 = 52
x = –2, r = 5
y=
Solve for y2
y=+
Because y > 0
We can now use the Double-Angle Formula for Sine.
Double-Angle Formula
From the triangle
Simplify
59

60.

Product-Sum FormulasIt is possible to write the product sin u cos as a sum of
trigonometric functions. To see this, consider the Addition
and Subtraction Formulas for Sine:
sin(u + ) = sin u cos + cos u sin
sin(u – ) = sin u cos – cos u sin
Adding the left- and right-hand sides of these formulas
gives
sin(u + ) = sin(u – ) = 2 sin u cos
60

61. Product-Sum Formulas

Dividing by 2 gives the formula
sin u cos = [sin(u + ) + sin(u – )]
The other three Product-to-Sum Formulas follow from the
Addition Formulas in a similar way.
61

62. Product-Sum Formulas

The Product-to-Sum Formulas can also be used as
Sum-to-Product Formulas. This is possible because the
right-hand side of each Product-to-Sum Formula is a sum
and the left side is a product. For example, if we let
in the first Product-to-Sum Formula, we get
so
62

63. Product-Sum Formulas

The remaining three of the following Sum-to-Product
Formulas are obtained in a similar manner.
63

64. Example 5 – Proving an Identity

Verify the identity
.
Solution:
We apply the second Sum-to-Product Formula to the
numerator and the third formula to the denominator.
Sum-to-Product
Formulas
64

65. Example 5 – Solution

cont’d
Simplify
Cancel
65
7.4
Basic Trigonometric Equations
66

67. Basic Trigonometric Equations

An equation that contains trigonometric functions is called a
trigonometric equation. For example, the following are
trigonometric equations:
sin2 + cos2 = 1
2 sin – 1 = 0
tan 2 – 1 = 0
The first equation is an identity—that is, it is true for every
value of the variable . The other two equations are true
only for certain values of .
To solve a trigonometric equation, we find all the values of
the variable that make the equation true.
67

68. Basic Trigonometric Equations

Solving any trigonometric equation always reduces to
solving a basic trigonometric equation—an equation of
the form T( ) = c, where T is a trigonometric function and c
is a constant.
In the next examples we solve such basic equations.
68

69. Example 1 – Solving a Basic Trigonometric Equation

Solve the equation
Solution:
Find the solutions in one period. Because sine has
period 2 , we first find the
solutions in any interval of
length 2 . To find these
solutions, we look at the
unit circle in Figure 1.
Figure 1
69

70. Example 1 – Solution

cont’d
We see that sin = in Quadrants I and II, so the solutions
in the interval [0, 2 ) are
Find all solutions. Because the sine function repeats its
values every 2 units, we get all solutions of the equation
by adding integer multiples of 2 to these solutions:
where k is any integer.
70

71. Example 1 – Solution

cont’d
Figure 2 gives a graphical representation of the solutions.
Figure 2
71

72. Example 2 – Solving a Basic Trigonometric Equation

Solve the equation tan = 2.
Solution:
Find the solutions in one period. We first find one
solution by taking tan–1 of each side of the equation.
tan = 2
Given equation
= tan–1(2)
Take tan–1 of each side
1.12
Calculator (in radian mode)
72

73. Example 2 – Solution

cont’d
By the definition of tan–1 the solution that we obtained is the
only solution in the interval (– /2, /2) (which is an interval
of length ).
Find all solutions. Since tangent has period , we get all
solutions of the equation by adding integer multiples of :
1.12 + k
where k is any integer.
73

74. Example 2 – Solution

cont’d
A graphical representation of the solutions is shown in
Figure 6.
Figure 6
You can check that the solutions shown in the graph
correspond to k = –1, 0, 1, 2, 3.
74

75. Basic Trigonometric Equations

In the next example we solve trigonometric equations that
are algebraically equivalent to basic trigonometric
equations.
75

76. Example 3 – Solving Trigonometric Equations

Find all solutions of the equation.
(a) 2 sin – 1 = 0
(b) tan2 – 3 = 0
Solution:
(a) We start by isolating sin .
2 sin – 1 = 0
2 sin = 1
sin =
Given equation
Add 1
Divide by 2
76

77. Example 3 – Solution

cont’d
This last equation is the same as that in Example 1. The
solutions are
=
+ 2k
=
+ 2k
where k is any integer.
(b) We start by isolating tan .
tan2 – 3 = 0
Given equation
tan2 = 3
Add 3
tan =
Take the square root
77

78. Example 3 – Solution

cont’d
Because tangent has period , we first find the solutions
in any interval of length . In the interval (– /2, /2) the
solutions are = /3 and = – /3.
To get all solutions, we add integer multiples of to
these solutions:
=
+ k
=–
+ k
where k is any integer.
78
Solving Trigonometric Equations
by Factoring
79

80. Solving Trigonometric Equations by Factoring

Factoring is one of the most useful techniques for solving
equations, including trigonometric equations.
The idea is to move all terms to one side of the equation,
factor, and then use the Zero-Product Property.
80

81. Example 4 – A Trigonometric Equation of Quadratic Type

Solve the equation 2 cos2 – 7 cos + 3 = 0.
Solution:
We factor the left-hand side of the equation.
2 cos2 – 7 cos + 3 = 0
(2 cos – 1)(cos – 3) = 0
2 cos – 1 = 0 or cos – 3 = 0
cos =
or
cos = 3
Given equation
Factor
Set each factor equal to 0
Solve for cos
81

82. Example 4 – Solution

cont’d
Because cosine has period 2 , we first find the solutions in
the interval [0, 2 ). For the first equation the solutions are
= /3 and = 5 /3 (see Figure 7).
Figure 7
82

83. Example 4 – Solution

cont’d
The second equation has no solution because cos is
never greater than 1.
Thus the solutions are
=
+ 2k
=
+ 2k
where k is any integer.
83

84. Example 5 – Solving a Trigonometric Equation by Factoring

Solve the equation 5 sin cos + 4 cos = 0.
Solution:
We factor the left-hand side of the equation.
5 sin cos + 2 cos = 0
cos (5 sin + 2) = 0
cos = 0
or 5 sin + 4 = 0
sin = –0.8
Given equation
Factor
Set each factor equal to 0
Solve for sin
84

85. Example 5 – Solution

cont’d
Because sine and cosine have period 2 , we first find the
solutions of these equations in an interval of length 2 .
For the first equation the solutions in the interval [0, 2 ) are
= /2 and = 3 /2 . To solve the second equation, we
take sin–1 of each side.
sin = –0.80
= sin–1(–0.80)
Second equation
Take sin–1 of each side
85

86. Example 5 – Solution

–0.93
cont’d
Calculator (in radian mode)
So the solutions in an interval of length 2 are = –0.93
and = + 0.93 4.07 (see Figure 8).
Figure 8
86

87. Example 5 – Solution

cont’d
We get all the solutions of the equation by adding integer
multiples of 2 to these solutions.
=
+ 2k
–0.93 + 2k
=
+ 2k
4.07 + 2k
where k is any integer.
87
7.5
More Trigonometric Equations
88

89. More Trigonometric Equations

In this section we solve trigonometric equations by first
using identities to simplify the equation. We also solve
trigonometric equations in which the terms contain
multiples of angles.
89
Solving Trigonometric Equations
by Using Identities
90

91. Solving Trigonometric Equations by Using Identities

In the next example we use trigonometric identities to
express a trigonometric equation in a form in which it can
be factored.
91

92. Example 1 – Using a Trigonometric Identity

Solve the equation 1 + sin = 2 cos2 .
Solution:
We first need to rewrite this equation so that it contains
only one trigonometric function. To do this, we use a
trigonometric identity:
1 + sin = 2 cos2
Given equation
1 + sin = 2(1 – sin2 )
Pythagorean identity
2 sin2 + sin – 1 = 0
(2 sin – 1)(sin + 1) = 0
Put all terms on one side
Factor
92

93.

Example 1 – Solution2 sin – 1 = 0
or
sin + 1 = 0
sin =
or
sin = –1
=
or
=
cont’d
Set each factor
equal to 0
Solve for sin
Solve for in the
interval [0, 2 )
Because sine has period 2 , we get all the solutions of the
equation by adding integer multiples of 2 to these
solutions.
93

94. Example 1 – Solution

cont’d
Thus the solutions are
=
+ 2k
=
+ 2k
=
+ 2k
where k is any integer.
94

95. Example 2 – Squaring and Using an Identity

Solve the equation cos + 1 = sin in the interval [0, 2 ).
Solution:
To get an equation that involves either sine only or cosine
only, we square both sides and use a Pythagorean identity.
cos + 1 = sin
Given equation
cos2 + 2 cos + 1 = sin2
Square both sides
cos2 + 2 cos + 1 = 1 – cos2
Pythagorean identity
2 cos2 + 2 cos = 0
Simplify
95

96. Example 2 – Solution

2 cos (cos + 1) = 0
cont’d
Factor
2 cos = 0
or
cos + 1 = 0
Set each factor
equal to 0
cos = 0
or
cos = –1
Solve for cos
=
or
=
Solve for in [0, 2 )
Because we squared both sides, we need to check for
extraneous solutions. From Check Your Answers we see
that the solutions of the given equation are /2 and .
96

97. Example 2 – Solution

cont’d
Check Your Answers:
=
cos
+ 1 = sin
0+1=1
=
cos
=
+ 1 = sin
cos + 1 = sin
0 + 1 ≟ –1
–1 + 1 = 0
97

98. Example 3 – Finding Intersection Points

Find the values of x for which the graphs of f(x) = sin x and
g(x) = cos x intersect.
Solution 1: Graphical
The graphs intersect where f(x) = g(x). In Figure 1 we
graph y1 = sin x and y2 = cos x on the same screen, for x
between 0 and 2 .
(a)
(b)
Figure 1
98

99. Example 3 – Solution

cont’d
Using
or the intersect command on the graphing
calculator, we see that the two points of intersection in this
interval occur where x 0.785 and x 3.927.
Since sine and cosine are periodic with period 2 , the
intersection points occur where
x 0.785 + 2k
and
x 3.927 + 2k
where k is any integer.
99

100. Example 3 – Solution

cont’d
Solution 2: Algebraic
To find the exact solution, we set f(x) = g(x) and solve the
resulting equation algebraically:
sin x = cos x
Equate functions
Since the numbers x for which cos x = 0 are not solutions
of the equation, we can divide both sides by cos x:
=1
tan x = 1
Divide by cos x
Reciprocal identity
100

101. Example 3 – Solution

cont’d
The only solution of this equation in the interval (– /2, /2)
is x = /4. Since tangent has period , we get all solutions
of the equation by adding integer multiples of :
x=
+ k
where k is any integer. The graphs intersect for these
values of x.
You should use your calculator to check that, rounded to
three decimals, these are the same values that we
obtained in Solution 1.
101

102. Equations with Trigonometric Functions of Multiples of Angles

When solving trigonometric equations that involve functions
of multiples of angles, we first solve for the multiple of the
angle, then divide to solve for the angle.
102

103. Example 4 – A Trigonometric Equation Involving a Multiple of an Angle

Consider the equation 2 sin 3 – 1 = 0.
(a) Find all solutions of the equation.
(b) Find the solutions in the interval [0, 2 ).
Solution:
(a) We first isolate sin 3 and then solve for the angle 3 .
2 sin 3 – 1 = 0
2 sin 3 = 1
sin 3 =
Given equation
Add 1
Divide by 2
103

104. Example 4 – Solution

3 =
cont’d
Solve for 3 in the interval
[0, 2 ) (see Figure 2)
Figure 2
104

105. Example 4 – Solution

cont’d
To get all solutions, we add integer multiples of 2 to
these solutions. So the solutions are of the form
3 =
+ 2k
3 =
+ 2k
To solve for , we divide by 3 to get the solutions
where k is any integer.
105

106. Example 4 – Solution

cont’d
(b) The solutions from part (a) that are in the interval [0, 2 )
correspond to k = 0, 1, and 2. For all other values of k
the corresponding values of lie outside this interval.
So the solutions in the interval [0, 2 ) are
106

107. Example 5 – A Trigonometric Equation Involving a Half Angle

Consider the equation
(a) Find all solutions of the equation.
(b) Find the solutions in the interval [0, 4 ).
Solution:
(a) We start by isolating tan
.
Given equation
Add 1
107

108. Example 5 – Solution

cont’d
Divide by
Solve for
in the interval
Since tangent has period , to get all solutions, we add
integer multiples of to this solution. So the solutions
are of the form
108

109. Example 5 – Solution

cont’d
Multiplying by 2, we get the solutions
=
+ 2k
where k is any integer.
(b) The solutions from part (a) that are in the interval
[0, 4 ) correspond to k = 0 and k = 1. For all other
values of k the corresponding values of x lie outside
this interval. Thus the solutions in the interval [0, 4 )
are
109

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тригонометрия — Показать, что $\sin(45^\circ+\alpha)=\cos(45^\circ-\alpha)$ и $\cos(45^\circ+\alpha)=\sin(45^\ круг-\альфа)$ 9\circ -\theta)$$ Диапазон $\alpha$ не имеет значения, так как это тождество верно для всех $\theta\in\mathbb R$.

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