Trigonometric Identities and 606 CHAPTER 7 Trigonometric Identities and Equations 7.1 Fundamental Identities

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  • 605

    7 Trigonometric Identities and Equations

    In 1831 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now we generate massive amounts of elec- tricity by simultaneously rotating thousands of wires near large electromagnets. Because electric current alternates its direction on electrical wires, it is modeled accurately by either the sine or the cosine function.

    We give many examples of applications of the trigonometric functions to electricity and other phenomena in the examples and exercises in this chapter, including a model of the wattage consumption of a toaster in Section 7.4, Example 5.

    7.1 Fundamental Identities

    7.2 Verifying Trigonometric Identities

    7.3 Sum and Difference Identities

    7.4 Double-Angle Identities and Half- Angle Identities

    Summary Exercises on Verifying Trigonometric Identities

    7.5 Inverse Circular Functions

    7.6 Trigonometric Equations

    7.7 Equations Involving Inverse Trigonometric Functions

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  • 606 CHAPTER 7 Trigonometric Identities and Equations

    7.1 Fundamental Identities Negative-Angle Identities ■ Fundamental Identities ■ Using the Fundamental Identities

    Figure 1

    TEACHING TIP Point out that in trigonometric identities, � can be an angle in degrees, a real num- ber, or a variable.

    Negative-Angle Identities As suggested by the circle shown in Figure 1, an angle � having the point on its terminal side has a corresponding angle

    with the point on its terminal side. From the definition of sine,

    (Section 5.2)

    so and are negatives of each other, or

    Figure 1 shows an angle � in quadrant II, but the same result holds for � in any quadrant. Also, by definition,

    (Section 5.2)

    so

    We can use these identities for and to find in terms of

    Similar reasoning gives the following identities.

    This group of identities is known as the negative-angle or negative-number identities.

    Fundamental Identities In Chapter 5 we used the definitions of the trigonometric functions to derive the reciprocal, quotient, and Pythagorean identities. Together with the negative-angle identities, these are called the fundamental identities.

    Fundamental Identities

    Reciprocal Identities

    Quotient Identities

    (continued)

    tan � � sin � cos �

    cot � � cos � sin �

    cot � � 1

    tan � sec � �

    1 cos �

    csc � � 1

    sin �

    csc���� � �csc �, sec���� � sec �, cot���� � �cot �

    tan���� � �tan �.

    tan���� � sin���� cos����

    � �sin �

    cos � � �

    sin �

    cos �

    tan �: tan����cos����sin����

    cos���� � cos �.

    cos���� � x

    r and cos � �

    x

    r ,

    sin���� � �sin �.

    sin �sin����

    sin���� � �y

    r and sin � �

    y

    r ,

    �x, �y��� �x, y�

    sin(–�) = – = –sin � y r

    O x

    y

    y x

    (x, y)

    –y

    r

    (x, –y)

    –�

    r

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  • 7.1 Fundamental Identities 607

    TEACHING TIP Encourage students to memorize the identities pre- sented in this section as well as subsequent sections. Point out that numerical values can be used to help check whether or not an identity was recalled correctly.

    Pythagorean Identities

    Negative-Angle Identities

    N O T E The most commonly recognized forms of the fundamental identities are given above. Throughout this chapter you must also recognize alternative forms of these identities. For example, two other forms of are

    Using the Fundamental Identities One way we use these identities is to find the values of other trigonometric functions from the value of a given trigono- metric function. Although we could find such values using a right triangle, this is a good way to practice using the fundamental identities.

    EXAMPLE 1 Finding Trigonometric Function Values Given One Value and the Quadrant

    If and � is in quadrant II, find each function value.

    (a) (b) (c)

    Solution

    (a) Look for an identity that relates tangent and secant.

    Pythagorean identity

    Combine terms.

    Take the negative square root. (Section 1.4)

    Simplify the radical. (Section R.7)

    We chose the negative square root since sec � is negative in quadrant II.

    � �34

    3 � sec �

    ��349 � sec � 34

    9 � sec2 �

    25

    9 � 1 � sec2 �

    tan � � �53 �� 53�2 � 1 � sec2 � tan2 � � 1 � sec2 �

    cot����sin �sec �

    tan � � �53

    sin2 � � 1 � cos2 � and cos2 � � 1 � sin2 �.

    sin2 � � cos2 � � 1

    csc(��) � �csc � sec(��) � sec � cot(��) � �cot �

    sin(��) � �sin � cos(��) � cos � tan(��) � �tan �

    sin2 � � cos2 � � 1 tan2 � � 1 � sec2 � 1 � cot2 � � csc2 �

    TEACHING TIP Warn students that the given information in

    Example 1,

    does not mean that and Ask them why these values cannot be correct.

    cos � � 3. sin � � �5

    tan � � � 5 3

    � �5 3

    ,

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  • 608 CHAPTER 7 Trigonometric Identities and Equations

    (b) Quotient identity

    Multiply by cos �.

    Reciprocal identity

    (c) Reciprocal identity

    Negative-angle identity

    Simplify. (Section R.5)

    Now try Exercises 5, 7, and 9.

    C A U T I O N To avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of � and the function being evaluated.

    Any trigonometric function of a number or angle can be expressed in terms of any other function.

    EXAMPLE 2 Expressing One Function in Terms of Another

    Express cos x in terms of tan x.

    Solution Since sec x is related to both cos x and tan x by identities, start with

    Take reciprocals.

    Reciprocal identity

    Take square roots.

    Quotient rule (Section R.7); rewrite.

    Rationalize the denominator. (Section R.7)

    Choose the � sign or the � sign, depending on the quadrant of x.

    Now try Exercise 43.

    cos x � ��1 � tan2 x

    1 � tan2 x

    cos x � �1

    �1 � tan2 x

    �� 11 � tan2 x � cos x

    1

    1 � tan2 x � cos2 x

    1

    1 � tan2 x �

    1

    sec2 x

    1 � tan2 x � sec2 x.

    tan � � � 53 ; cot���� � 1

    ���53� �

    3

    5

    cot���� � 1

    �tan �

    cot���� � 1

    tan����

    sin � � 5�34

    34

    ��3�3434 ��� 53� � sin � � 1sec ��tan � � sin �

    cos � tan � � sin �

    tan � � sin �

    cos �

    From part (a),

    tan � � �53

    1 sec � � �

    3 �34 � �

    3�34 34 ;

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  • 7.1 Fundamental Identities 609

    Figure 2 Write each fraction with the LCD. (Section R.5)

    Concept Check Fill in the blanks.

    1. If , then .

    2. If , then .

    3. If , then .

    4. If and , then .

    Find sin s. See Example 1.

    5. , s in quadrant I 6. , s in quadrant IV

    7. , 8. ,

    9. , 10.

    11. Why is it unnecessary to give the quadrant of s in Exercise 10?

    csc s � � 8

    5 tan s � 0sec s �

    11

    4

    sec s � 0tan s � � �7

    2 tan s � 0cos��s� �

    �5 5

    cot s � � 1

    3 cos s �

    3

    4

    tan��x� �sin x � .6cos x � .8 cot x �tan x � 1.6

    cos��x� �cos x � �.65 tan��x� �tan x � 2.6

    1. �2.6 2. �.65 3. .625

    4. �.75 5. 6.

    7. 8.

    9. 10. � 5

    8 �

    �105 11

    � �77 11

    � 2�5

    5

    � 3�10

    10 �7

    4

    7.1 Exercises

    9

    4

    –4

    –2� 2�

    Y1 = Y2

    4

    –4

    –2� 2�

    The graph supports the result in Example 3. The graphs of y1 and y2 appear to be iden- tical.

    y1 = tan x + cot x

    y2 = cos x sin x 1

    We can use a graphing calculator to decide whether two functions are identical. See Figure 2, which supports the identity With an identity, you should see no difference in the two graphs. ■

    All other trigonometric functions can easily be expressed in terms of and/or We often make such substitutions in an expression to simplify it.

    EXAMPLE 3 Rewriting an Expression in Terms of Sine and Cosine

    Write in terms of and and then simplify the expression.

    Solution

    Quotient identities

    Add fractions.

    Pythagorean identity

    Now try Exercise 55.

    C A U T I O N When working with trigonometric expressions and identities, be sure to write the argument of the function. For example, we