1. 7 Trigonometric Identities and Equations
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2. Trigonometric Identities and Equations7
7.1 Fundamental Identities
7.2 Verifying Trigonometric Identities
7.3 Sum and Difference Identities
7.4 Double-Angle and Half-Angle Identities
7.5 Inverse Circular Functions
7.6 Trigonometric Equations
7.7 Equations Involving Inverse Trigonometric Functions
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3. Sum and Difference Identities
7.3
Sum and Difference Identities
Cosine Sum and Difference Identities ▪ Cofunction Identities ▪ Sine and Tangent Sum and Difference Identities
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4. 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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5. 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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6. Difference Identity for Cosine
Square both sides and clear parentheses:
Rearrange the terms:
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7. Difference Identity for Cosine
Subtract 2, then divide by –2:
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8. 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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9. Cosine of a Sum or Difference
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10. 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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11. Example 1(b) Find the exact value of
FINDING EXACT COSINE FUNCTION VALUES
Find the exact value of
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12. 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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13. Cofunction Identities
Similar identities can be obtained for a real number domain by replacing 90 with
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14. 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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15. 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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16. 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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17. 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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18. Write cos(180° – θ) as a trigonometric function of θ alone.
Example 3
REDUCING cos (A – B) TO A FUNCTION OF A SINGLE VARIABLE
Write cos(180° – θ) as a trigonometric function of θ alone.
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19. 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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20. Sum and Difference Identities for Sine
Sine sum identity
Negative-angle identities
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21. Sine of a Sum or Difference
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22. Sum and Difference Identities for Tangent
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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23. Sum and Difference Identities for Tangent
Multiply.
Simplify.
Fundamental identity
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24. Sum and Difference Identities for Tangent
Replace B with –B and use the fact that tan(–B) = –tan B to obtain the identity for the tangent of the difference of two angles.
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25. Tangent of a Sum or Difference
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26. Find the exact value of sin 75.
Example 4(a)
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of sin 75.
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27. Example 4(b) Find the exact value of
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of
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28. Example 4(c) Find the exact value of
FINDING EXACT SINE AND TANGENT FUNCTION VALUES
Find the exact value of
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29. Write each function as an expression involving functions of θ.
Example 5
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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30. Example 6
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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31. Because A is in quadrant II, cos A is negative and tan A is negative.
Example 6
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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32. Because B is in quadrant III, sin B is negative and tan B is positive.
Example 6
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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33. Example 6
FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)
(a)
(b)
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34. From parts (a) and (b), sin (A + B) > 0 and tan (A + B) > 0.
Example 6
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 I.
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35. Example 7
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  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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36. Example 7
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  for these electric generators.
Each cycle is 2 radians at 60 cycles per second, so the angular speed is  = 60(2) = 120 radians per second.
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37. (b) Graph V in the window [0, .05] by [–200, 200].
Example 7
APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued)
(b) Graph V in the window [0, .05] by [–200, 200].
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38. Example 7
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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