1. Fundamental Identities
Reciprocal Identities
Quotient Identities
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2. Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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3. If and θ is in quadrant II, find each function value.
Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
If and θ is in quadrant II, find each function value.
(a) sec θ
Pythagorean identity
In quadrant II, sec θ is negative, so
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4. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
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5. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) cot(– θ)
Reciprocal identity
Negative-angle identity
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6. Express cos x in terms of tan x.
Example 2
EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER
Express cos x in terms of tan x.
Since sec x is related to both cos x and tan x by identities, start with
Take reciprocals.
Reciprocal identity
Take the square root of each side.
The sign depends on the quadrant of x.
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7. Example 3
REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE
Write tan θ + cot θ in terms of sin θ and cos θ, and then simplify the expression.
Quotient identities
Write each fraction with the LCD.
Pythagorean identity
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8. Verify that is an identity.
Example 1
VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)
Verify that is an identity.
Work with the right side since it is more complicated.
Right side of given equation
Distributive property
Left side of given equation
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9. Verify that is an identity.
Example 2
VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)
Verify that is an identity.
Left side
Distributive property
Right side
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10. Verify that is an identity.
Example 3
VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)
Verify that is an identity.
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11. Verify that is an identity.
Example 4
VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)
Verify that is an identity.
Multiply by 1 in the form
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12. Verifying Identities by Working with Both Sides
If both sides of an identity appear to be equally complex, the identity can be verified by working independently on each side until they are changed into a common third result.
Each step, on each side, must be reversible.
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13. Verify that is an identity.
Example 5
VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES)
Verify that is an identity.
Working with the left side:
Multiply by 1 in the form
Distributive property
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14. Example 5 Working with the right side:
VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) (continued)
Working with the right side:
Factor the numerator.
Factor the denominator.
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15. So, the identity is verified.
Example 5
VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) (continued)
Right side of given equation
Left side of given equation
Common third expression
So, the identity is verified.
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16. Sum and Difference Identitites for Cosine
5.3
Sum and Difference Identitites for Cosine
Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities
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17. Difference Identity for Cosine
Point Q is on the unit circle, so the coordinates of Q are (cos B, sin B).
The coordinates of S are (cos A, sin A).
The coordinates of R are (cos(A – B), sin (A – B)).
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18. Difference Identity for Cosine
Since the central angles SOQ and POR are equal, PR = SQ.
Using the distance formula, we have
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19. Difference Identity for Cosine
Square both sides and clear parentheses:
Rearrange the terms:
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20. Difference Identity for Cosine
Subtract 2, then divide by –2:
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21. Sum Identity for Cosine
To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).
Cosine difference identity
Negative angle identities
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22. Cosine of a Sum or Difference
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23. Find the exact value of cos 15.
Example 1(a)
FINDING EXACT COSINE FUNCTION VALUES
Find the exact value of cos 15.
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24. Example 1(b) Find the exact value of
FINDING EXACT COSINE FUNCTION VALUES
Find the exact value of
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25. Find the exact value of cos 87cos 93 – sin 87sin 93.
Example 1(c)
FINDING EXACT COSINE FUNCTION VALUES
Find the exact value of cos 87cos 93 – sin 87sin 93.
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26. Cofunction Identities
Similar identities can be obtained for a real number domain by replacing 90 with
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27. Find an angle that satisfies each of the following:
Example 2
USING COFUNCTION IDENTITIES TO FIND θ
Find an angle that satisfies each of the following:
(a) cot θ = tan 25°
(b) sin θ = cos (–30°)
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28. Find an angle that satisfies each of the following:
Example 2
USING COFUNCTION IDENTITIES TO FIND θ
Find an angle that satisfies each of the following:
(c)
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29. Note
Because trigonometric (circular) functions are periodic, the solutions in Example 2 are not unique. Only one of infinitely many possiblities are given.
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30. Applying the Sum and Difference Identities
If one of the angles A or B in the identities for cos(A + B) and cos(A – B) is a quadrantal angle, then the identity allows us to write the expression in terms of a single function of A or B.
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31. Write cos(90° + θ) as a trigonometric function of θ alone.
Example 3
REDUCING cos (A – B) TO A FUNCTION OF A SINGLE VARIABLE
Write cos(90° + θ) as a trigonometric function of θ alone.
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32. Suppose that and both s and t are in quadrant II. Find cos(s + t).
Example 4
FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t
Suppose that and both s and t are in quadrant II. Find cos(s + t).
Sketch an angle s in quadrant II such that Since
let y = 3 and r = 5.
The Pythagorean theorem gives
Since s is in quadrant II, x = –4 and
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33. Sketch an angle t in quadrant II such that Since
Example 4
FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)
Sketch an angle t in quadrant II such that Since
let x = –12 and r = 5.
The Pythagorean theorem gives
Since t is in quadrant II, y = 5 and
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34. Example 4 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)
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35. Note
The values of cos s and sin t could also be found by using the Pythagorean identities.
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36. Example 5
APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE
Common household current is called alternating current because the current alternates direction within the wires. The voltage V in a typical 115-volt outlet can be expressed by the function
where ω is the angular speed (in radians per second) of the rotating generator at the electrical plant, and t is time measured in seconds.*
*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 1988.)
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37. Example 5
APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued)
(a) It is essential for electric generators to rotate at precisely 60 cycles per second so household appliances and computers will function properly. Determine ω for these electric generators.
Each cycle is 2π radians at 60 cycles per second, so the angular speed is ω = 60(2π) = 120π radians per second.
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38. (b) Graph V in the window [0, .05] by [–200, 200].
Example 5
APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued)
(b) Graph V in the window [0, .05] by [–200, 200].
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39. Example 5
APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued)
(c) Determine a value of so that the graph of is the same as the graph of
Using the negative-angle identity for cosine and a cofunction identity gives
Therefore, if
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40. Sum and Difference Identities for Sine and Tangent
5.4
Sum and Difference Identities for Sine and Tangent
Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities
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41. Sum and Difference Identities for Sine
We can use the cosine sum and difference identities to derive similar identities for sine and tangent.
Cofunction identity
Cosine difference identity
Cofunction identities
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42. Sum and Difference Identities for Sine
Sine sum identity
Negative-angle identities
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43. Sine of a Sum or Difference
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44. Sum and Difference Identities for Tangent
We can use the cosine sum and difference identities to derive similar identities for sine and tangent.
Fundamental identity
Sum identities
Multiply numerator and denominator by 1.
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45. Sum and Difference Identities for Tangent
Multiply.
Simplify.
Fundamental identity
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46. Sum and Difference Identities for Tangent
Replace B with –B and use the fact that tan(–B) to obtain the identity for the tangent of the difference of two angles.
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47. Tangent of a Sum or Difference
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48. Find the exact value of sin 75.
Example 1(a)
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of sin 75.
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49. Example 1(b) Find the exact value of
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of
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50. Example 1(c) Find the exact value of
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of
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51. Write each function as an expression involving functions of θ.
Example 2
WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF θ
Write each function as an expression involving functions of θ.
(a)
(b)
(c)
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52. Example 3
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B
Suppose that A and B are angles in standard position with Find each of the following.
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53. Because A is in quadrant II, cos A is negative and tan A is negative.
Example 3
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)
The identity for sin(A + B) requires sin A, cos A, sin B, and cos B. The identity for tan(A + B) requires tan A and tan B. We must find cos A, tan A, sin B and tan B.
Because A is in quadrant II, cos A is negative and tan A is negative.
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54. Because B is in quadrant III, sin B is negative and tan B is positive.
Example 3
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)
Because B is in quadrant III, sin B is negative and tan B is positive.
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55. Example 3
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)
(a)
(b)
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56. From parts (a) and (b), sin (A + B) > 0 and tan (A − B) > 0.
Example 3
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)
From parts (a) and (b), sin (A + B) > 0 and tan (A − B) > 0.
The only quadrant in which the values of both the sine and the tangent are positive is quadrant I, so (A + B) is in quadrant IV.
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57. Verify that the equation is an identity.
Example 4
VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES
Verify that the equation is an identity.
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58. Double-Angle Identities
5.5
Double-Angle Identities
Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities
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59. Double-Angle Identities
We can use the cosine sum identity to derive double-angle identities for cosine.
Cosine sum identity
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60. Double-Angle Identities
There are two alternate forms of this identity.
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61. Double-Angle Identities
We can use the sine sum identity to derive a double-angle identity for sine.
Sine sum identity
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62. Double-Angle Identities
We can use the tangent sum identity to derive a double-angle identity for tangent.
Tangent sum identity
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63. Double-Angle Identities
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64. Any of the three forms may be used.
Example 1
FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ
Given and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ.
The identity for sin 2θ requires sin θ.
Any of the three forms may be used.
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65. Now find tan θ and then use the tangent double-angle identity.
Example 1
FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.)
Now find tan θ and then use the tangent double-angle identity.
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66. Example 1
FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.)
Alternatively, find tan 2θ by finding the quotient of sin 2θ and cos 2θ.
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67. Find the values of the six trigonometric functions of θ if
Example 2
FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ
Find the values of the six trigonometric functions of θ if
Use the identity to find sin θ:
θ is in quadrant II, so sin θ is positive.
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68. Example 2
FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ (cont.)
Use a right triangle in quadrant II to find the values of cos θ and tan θ.
Use the Pythagorean theorem to find x.
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69. Verify that is an identity.
Example 3
VERIFYING A DOUBLE-ANGLE IDENTITY
Verify that is an identity.
Quotient identity
Double-angle identity
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70. Simplify each expression.
Example 4
SIMPLIFYING EXPRESSION DOUBLE-ANGLE IDENTITIES
Simplify each expression.
Multiply by 1.
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71. Write sin 3x in terms of sin x.
Example 5
DERIVING A MULTIPLE-ANGLE IDENTITY
Write sin 3x in terms of sin x.
Sine sum identity
Double-angle identities
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72. Example 6
DETERMINING WATTAGE CONSUMPTION
If a toaster is plugged into a common household outlet, the wattage consumed is not constant. Instead it varies at a high frequency according to the model
where V is the voltage and R is a constant that measure the resistance of the toaster in ohms.*
Graph the wattage W consumed by a typical toaster with R = 15 and in the window [0, .05] by [–500, 2000]. How many oscillations are there?
*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 1988.)
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73. There are six oscillations.
Example 6
DETERMINING WATTAGE CONSUMPTION
There are six oscillations.
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74. Product-to-Sum Identities
The identities for cos(A + B) and cos(A – B) can be added to derive a product-to-sum identity for cosines.
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75. Product-to-Sum Identities
Similarly, subtracting cos(A + B) from cos(A – B) gives a product-to-sum identity for sines.
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76. Product-to-Sum Identities
Using the identities for sin(A + B) and sine(A – B) gives the following product-to-sum identities.
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77. Product-to-Sum Identities
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78. Write 4 cos 75° sin 25° as the sum or difference of two functions.
Example 7
USING A PRODUCT-TO-SUM IDENTITY
Write 4 cos 75° sin 25° as the sum or difference of two functions.
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79. Sum-to-Product Identities
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80. Write as a product of two functions.
Example 8
USING A SUM-TO-PRODUCT IDENTITY
Write as a product of two functions.
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81. Half-Angle Identities
5.6
Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities
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82. Half-Angle Identities
We can use the cosine sum identities to derive half-angle identities.
Choose the appropriate sign depending on the quadrant of
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83. Half-Angle Identities
Choose the appropriate sign depending on the quadrant of
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84. Half-Angle Identities
There are three alternative forms for
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85. Half-Angle Identities
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86. Double-Angle Identities
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87. Example 1
USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE
Find the exact value of cos 15° using the half-angle identity for cosine.
Choose the positive square root.
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88. Find the exact value of tan 22.5° using the identity
Example 2
USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE
Find the exact value of tan 22.5° using the identity
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89. The angle associated with lies in quadrant II since
Example 3
FINDING FUNCTION VALUES OF s/2 GIVEN INFORMATION ABOUT s
The angle associated with lies in quadrant II since
is positive while are negative.
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90. Example 3
FINDING FUNCTION VALUES OF s/2 GIVEN INFORMATION ABOUT s (cont.)
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91. Simplify each expression.
Example 4
SIMPLIFYING EXPRESSIONS USING THE HALF-ANGLE IDENTITIES
Simplify each expression.
This matches part of the identity for
Substitute 12x for A:
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92. Verify that is an identity.
Example 5
VERIFYING AN IDENTITY
Verify that is an identity.
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