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Fig. 7.10
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CHAPTER 7 Graphs of the Trigonometric Functions
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(g) y
tan 2x
Fig. 7.10 (Cont.)
Construct the graph of each of the following for one period (see Fig. 7.11). (a) y (b) y sin x 2 cos x sin 3x cos 2x (c) y (d) y sin 2x sin 3x sin 3x cos 2x
CHAPTER 7 Graphs of the Trigonometric Functions
Ans.
(a) y
sin x
2 cos x
(b) y
sin 3x
cos 2x
(c) y
sin 2x
sin 3x
(d) y
sin 3x
cos 2x
Fig. 7.11
Construct the graph of each of the following for one period (see Fig. 7.12). (a) y (b) y Ans. sin x cos x (a) y 3 2 sin x 3 (c) y (d) y sin (x cos (x /4) /6) (b) y cos x 2
Fig. 7.12
CHAPTER 7 Graphs of the Trigonometric Functions
(c) y
sin (x
(d) y
cos (x
Fig. 7.12 (Cont.)
The horizontal displacement, d, of the bob on a pendulum is given by d a sin bt, where d is expressed in centimeters, t is in seconds, and a and b are constants. Find the amplitude and period of the displacement of the bob on a pendulum given by: (a) d (b) d Ans. 10 sin 2 t 12 sin 4 t (a) Amplitude 10 cm, period 1 s (b) Amplitude 12 cm, period 0.5 s
The voltage E in an electric circuit is given by E a cos bt, where a and b are constants and t is the time measured in seconds. Find the amplitude and period of the voltage given by: (a) E (b) E Ans. 3.6 cos 80 t 6.2 cos 20 t (a) Amplitude 3.6 V, period 0.025 s (b) Amplitude 6.2 V, period 0.1 s
The pressure, P, in a traveling sound wave is given by P a sin b(t c), where a, b, and c are constants, P is the pressure in dynes per square centimeter, and t is in seconds. Find the amplitude, period, and phase shift of the pressure given by: (a) P (b) P Ans. 20 sin 100 (t 40 sin 200 (t 0.2) 0.5)
(a) Amplitude 20 dyn/cm2, period 0.02 s, phase shift 0.2 s (b) Amplitude 40 dyn/cm2, period 0.01 s, phase shift 0.5 s
Basic Relationships and Identities
8.1 Basic Relationships
Reciprocal Relationships csc u sec u cot u 1 sin u 1 cos u 1 tan u Quotient Relationships tan u cot u sin u cos u cos u sin u Pythagorean Relationships sin2 u cos2 u 1
tan2 u cot2 u
sec2 u csc2 u
The basic relationships hold for every value of for which the functions involved are defined. Thus, sin2 cos2 1 holds for every value of , while tan sin /cos holds for all values of for which tan is defined, i.e., for all n 90 where n is odd. Note that for the excluded values of , cos 0 and sin 0. For proofs of the quotient and Pythagorean relationships, see Probs. 8.1 and 8.2. The reciprocal relationships were treated in Chap. 2. (See also Probs. 8.3 to 8.6.)
8.2 Simplification of Trigonometric Expressions
It is frequently desirable to transform or reduce a given expression involving trigonometric functions to a simpler form.
EXAMPLE 8.1 (a) Using csc u
(b) Using tan u
1 , cos u csc u sin u sin u , cos u tan u cos u cos2 u
1 sin u sin u cos u cos u cos u
cos u sin u sin u.
cot u.
EXAMPLE 8.2 Using the relation sin2 u
1, (1) sin u 1 sin u. sin u.
(a) sin u
sin u cos u
(sin u
cos u) sin u
cos 2 u sin u
sin 2 u sin u
sin u)(1 sin u) 1 sin u
CHAPTER 8 Basic Relationships and Identities
(NOTE: The relation sin2 u cos2 u 1 may be written as sin2 u 1 is equally useful. In Example 8.2 the second of these forms was used.) cos2 u and as cos2 u
sin2 u. Each form
(See Probs. 8.7 to 8.9.)
8.3 Trigonometric Identities
An equation involving the trigonometric functions which is valid for all values of the angle for which the functions are defined is called a trigonometric identity. The eight basic relationships in Sec. 8.1 are trigonometric identities; so too are cos csc cot and cos tan sin
of Example 8.1. A trigonometric identity is verified by transforming one member (your choice) into the other. In general, one begins with the more complicated side. In some cases each side is transformed into the same new form.
General Guidelines for Verifying Identities
1. Know the eight basic relationships and recognize alternative forms of each. 2. Know the procedures for adding and subtracting fractions, reducing fractions, and transforming fractions into equivalent fractions. 3. Know factoring and special product techniques. 4. Use only substitution and simplification procedures that allow you to work on exactly one side of an equation. 5. Select the side of the equation that appears more complicated and attempt to transform it into the form of the other side of the equation. (See Example 8.3.) 6. If neither side is uncomplicated, transform each side of the equation, independently, into the same form. (See Example 8.4.) 7. Avoid substitutions that introduce radicals. 8. Use substitutions to change all trigonometric functions into expressions involving only sine and cosine and then simplify. (See Example 8.5.) 9. Multiply the numerator and denominator of a fraction by the conjugate of either. (See Example 8.6.) 10. Simplify a square root of a fraction by using conjugates to transform it into the quotient of perfect squares. (See Example 8.7.)
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