slide 6.1 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
Slide 6.1 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
OBJECTIVES
Trigonometric Identities and Equations
Learn to use the fundamental trigonometric identities to evaluate trigonometric functions.Learn to simplify a complicated trigonometric expression.Learn to verify a trigonometric identity.
SECTION 6.1
1
2
3
Slide 6.1 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
1. Reciprocal Identities
csc x 1
sin xsec x
1
cos xcot x
1
tan x
sin x 1
csc xcos x
1
sec xtan x
1
cot x
2. Quotient Identities
tan x sin x
cos xcot x
cos x
sin x
Slide 6.1 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
3. Pythagorean Identities
sin2 x cos2 x 1
4. Even - Odd Identities
sin x sin x
1 tan2 x sec2 x
1 cot2 x csc2 x
cos x cos x
tan x tan x
csc x csc x
sec x sec x
cot x cot x
Slide 6.1 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using the Fundamental Trigonometric Identities
If find the values
of the remaining trigonometric functions.
cot x 3
4 and
32
,
Solution
csc2 x 1 cot2 x
csc2 x 13
4
2
csc2 x 19
16
csc2 x 25
16
csc x 5
4
sin x 4
5
x is in QIIIcsc is negative
sin x 1
csc x
Slide 6.1 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2Using the Fundamental Trigonometric Identities
cos x
sin xcot x
csc x 5
4
sin x 4
5
Solution continued
sin xcos x
sin xcot x sin x
cos x cot x sin x
cos x 3
4
4
5
cos x 3
5
cos x 3
5
sec x 5
3cot x
3
4
tan x 4
3
Slide 6.1 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3Simplifying by Expressing All Trigonometric Functions in Terms of Sines and Cosines
Write in terms of sines and cosines and then simplify the resulting expression.
cot x tan x csc x sec x
Solutioncot x tan x csc x sec x
cos x
sin x
sin x
cos x
1
sin x
1
cos x
cos2 x
sin x cos x
sin2 x
sin x cos x
1
sin x cos x
Slide 6.1 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3Simplifying by Expressing All Trigonometric Functions in Terms of Sines and Cosines
Solution continued
cos2 x sin2 x 1
sin x cos x
11
sin x cos x
2
sin x cos x
Slide 6.1 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
TRIGONOMETRIC EQUATIONSAND IDENTITIES
To verify that a trigonometric equation is an identity, you must prove that both sides of the equation are equal for all values of the variable for which both sides are defined.
To prove that a trigonometric equation is NOT an identity, you find at least one value of the variable for which both sides are defined, but for which the two sides of the equation have different values.
Slide 6.1 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
VERIFYING TRIGONEMTIC IDENTITES
To verify that a given equation is an identity, we transform one side of the equation into the other side by a sequence of steps, each of which produces an identity. The steps involved may be algebraic manipulations or may use known identities.
Note that in verifying an identity, we do not just perform the same operation on both sides of the equation.
Slide 6.1 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Guidelines for Verifying Trigonometric IdentitiesAlgebra OperationsReview the use of the algebraic operations for combining fractions by finding the least common denominator. Fundamental Trigonometric Identities Review the fundamental trigonometric identities. Look for an opportunity to apply the fundamental trigonometric identities when working on either side of the identity to be verified. Be thoroughly familiar with alternative forms of fundamental identities.
Slide 6.1 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Guidelines for Verifying Trigonometric Identities
1. Start with the more complicated side. If one side of an identity is more complex than the other side, it is generally helpful to start with the more complicated side and simplify it until it becomes identical to the other side.
2. Stay focused on the answer. While working on one side of the identity, stay focused on your goal of converting it to the form on the other side.
Slide 6.1 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Guidelines for Verifying Trigonometric Identities
3. Convert to sines and cosines. Writing one side of the identity in terms of sines and cosines is often helpful.
4. Work on both sides. Sometimes, it is helpful to work separately on both sides of the equation to transform each side to the same equivalent expression.
Slide 6.1 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Guidelines for Verifying Trigonometric Identities
5. Use conjugates. In expressions containing 1 + sin x, 1 – sin x, 1 + cos x, 1 – cos x, sec x + tan x, and so on, it is often helpful to multiply both the numerator and the denominator by the appropriate conjugate and then use one of the forms of the Pythagorean identities.
Slide 6.1 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7Verifying by Rewriting with Sines and Cosines
Verify the identity: cot4 x cot2 x cot2 x csc2 x
cot4 x cot2 x cos4 x
sin4 x
cos2 x
sin2 x
Solution
Start with the more complicated left side.
cos4 x
sin4 x
cos2 x
sin2 xsin2 x
sin2 x
cos4 x cos2 x sin2 x
sin4 x
Slide 6.1 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7Verifying by Rewriting with Sines and Cosines
Solution continued
Because the left side is identical to the right side, the given equation is an identity.
cos2 x cos2 x sin2 x
sin4 x
cos2 x 1
sin4 x
cos2 x
sin2 x
1
sin2 x
cot2 x csc2 x
Slide 6.1 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7Verifying by Rewriting with Sines and Cosines
Solution continued
Rewriting expressions using only sines and cosines is not necessarily the quickest way to verify an identity, but it may help if you are stuck.
Here’s another way to verify this identity.
cot4 x cot2 x cot2 x cot2 x 1
cot4 x cot2 x cot2 x csc2 x
cot4 x cot2 x cot2 x csc2 x
Factor the left side.
Slide 6.1 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9 Verifying an Identity by Using a Conjugate
Verify the identity:cos x
1 sin x
1 sin x
cos x
cos x
1 sin x
cos x 1 sin x 1 sin x 1 sin x
Solution
Start with the left side.
cos x 1 sin x
1 sin2 x
cos x 1 sin x
cos2 x
1 sin x cos x