LHS Trig 8th ed

Ch 5 Notes F07 O’Brien

Trigonometry Chapter 5 Lecture Notes Section 5.1 I.

Fundamental Identities

Negative-Angle Identities sin (– θ) = – sin θ

csc (– θ) = – csc θ

tan (– θ) = – tan θ

cot (– θ) = – cot θ

cos (– θ) = cos θ

sec (– θ) = sec θ

One of the easiest ways to remember the negative-angle identities is to remember that only cosine and its reciprocal, secant are even functions. For even functions, f(– x) = f(x) which means these functions have y-axis symmetry. The other four trig functions (sine, cosecant, tangent, and cotangent) are odd functions. For odd functions, f(– x) = – f(x) which means these functions have origin symmetry. Example 1 Since tangent is an odd function, if tan x = 2.6, then tan (–x) = –2.6. (#1) II.

Reciprocal Identities

csc θ = III.

1 sin θ

sec θ =

1 cos θ

cot θ =

cos θ sin θ

cot θ =

1 tan θ

Quotient Identities

tan θ =

sin θ cos θ

IV. Pythagorean Identities sin 2 θ + cos 2 θ = 1 V.

tan 2 θ + 1 = sec 2 θ

1 + cot 2 θ = csc 2 θ

Using the Fundamental Identities A.

Finding Trigonometric Function Values Given One Value and the Quadrant

Example 2 Given cot x = −

1 and x is in quadrant IV, find sin x. (modified #6) 3

csc 2 x = 1 + cot 2 x → csc x = ± 1 + cot 2 x ; cosecant and sine are negative in IV; 2

1 3 3 10 10 10 ⎛ 1⎞ =− =− → sin x = csc x = − 1 + ⎜ − ⎟ = − =− 9 3 csc x 10 ⎝ 3⎠ 10 Now find the three remaining trigonometric functions of x. 1 tan x = = −3 cot x 1 3 10 10 = cos x = cot x ⋅ sin x → cos x = − ⋅ − 3 10 10 1 sec x = = 10 cos x 1

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B.

Ch 5 Notes F07 O’Brien

Using Identities to Rewrite Functions and Expressions

Example 3 Use identities to rewrite cot x in terms of sin x. (#44)

cot x =

cos x and from sin 2 x + cos 2 x = 1 we know cos x = ± 1 − sin 2 x sin x

therefore, cot x =

± 1 − sin 2 x . sin x

Example 4 Rewrite the expression sec θ ⋅ cot θ ⋅ sin θ in terms of sine and cosine and simplify. (#50)

sec θ ⋅ cot θ ⋅ sin θ =

1 cos θ cos θ ⋅ sin θ ⋅ ⋅ sin θ = =1 cos θ sin θ cos θ ⋅ sin θ

Example 5 Use identities to rewrite the expression sin2x + tan2x + cos2x in terms of sec x. (#63) sin2x + cos2x = 1, so sin2x + tan2x + cos2x = 1 + tan2x which equals sec2x ************************************************************************************

Section 5.2 I.

II.

Verifying Trigonometric Identities

Verifying Trigonometric Identities A.

An identity is an equation that is true for all of its domain values.

B.

To verify an identity, we show that one side of the identity can be rewritten to look exactly like the other side.

C.

Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same term to both sides or multiplying both sides by the same factor, are not valid when verifying identities.

Hints for Verifying Trigonometric Identities A.

Know the fundamental identities and their equivalent forms inside out and upside down.

Example 1

sin2x + cos2x = 1 is equivalent to cos2x = 1 – sin2x

Example 2

tan x =

sin x is equivalent to sin x = cos x ⋅ tan x cos x

B.

Start working with the more complicated side of the identity and try to turn it into the simpler side. Do not work on both sides of the identity simultaneously.

C.

Perform any indicated operations such as factoring, squaring binomials, distributing, or adding fractions.

Example 3

2 sin 2 x + 3 sin x + 1 can be factored to (2 sin x + 1)(sin x + 1) (#17)

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Example 4

D.

Ch 5 Notes F07 O’Brien

1 1 + can be added by getting a common denominator sin x cos x 1 1 1 cos x 1 sin x cos x + sin x + = ⋅ + ⋅ = sin x cos x sin x cos x cos x sin x sin x ⋅ cos x

Sometimes it is helpful to express all trigonometric functions on one side of an identity in terms of sine and cosine. tan x Example 5 Verify = sin x (#34) sec x sin x tan x cos x sin x cos x = = ⋅ = sin x 1 sec x cos x 1 cos x

E.

Fractions with a sum in the numerator and a single term in the denominator can be rewritten as the sum of two fractions.

Example 6

Verify

1 + cos x = csc x + cot x sin x

1 + cos x 1 cos x = + = csc x + cot x sin x sin x sin x Fractions with a difference in the numerator and a single term in the denominator can be rewritten as the difference of two fractions. F.

Sometimes it is helpful to rewrite one side of the identity in terms of a single trigonometric function. sin 2 x Example 7 Verify = sec x − cos x (#42) cos x

sin 2 x 1 − cos 2 x 1 cos 2 x = = − = sec x − cos x cos x cos x cos x cos x G.

Multiplying both the numerator and denominator of a fraction by the same factor (usually the conjugate of the numerator or denominator) may yield a Pythagorean identity and bring you closer to your goal.

Example 8

Verify

1 1 − sin x = . 1 + sin x cos 2 x

1 1 1 − sin x 1 − sin x 1 − sin x = ⋅ = = 1 + sin x 1 + sin x 1 − sin x 1 − sin 2 x cos 2 x

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H.

Ch 5 Notes F07 O’Brien

As you selection substitutions, keep in mind the side you are not changing. It represents your goal. Look for the identity or function which best links the two sides.

(

(

)

Verify tan 2 x 1 + cot 2 x =

Example 9

1 1 − sin 2 x

.

)

1 ⎞ 1 1 ⎛ 2 2 tan 2 x 1 + cot 2 x = tan 2 x⎜1 + = tan x + 1 = sec x = = ⎟ cos 2 x 1 − sin 2 x ⎝ tan 2 x ⎠ I.

If you get really stuck, abandon the side you’re working on and start working on the other side. Try to make the two sides “meet in the middle.”

(sec x − tan x )2 = 1 − sin x

Example 10

1 + sin x

working on left side:

working on right side:________

(sec x − tan x )2 =

1 − sin x = 1 + sin x

sec 2 x − 2 sec x tan x + tan 2 x =

1 − sin x 1 − sin x ⋅ = 1 + sin x 1 − sin x

1 cos 2 x 1 cos 2 x

−2 −

1 sin x sin 2 x ⋅ + = cos x cos x cos 2 x

2 sin x cos 2 x

+

1 − 2 sin x + sin 2 x 1 − sin 2 x

sin 2 x

1 − 2 sin x + sin 2 x

cos 2 x

cos 2 x 1 cos 2 x

−

2 sin x cos 2 x

+

= = sin 2 x cos 2 x

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Section 5.3 I.

Sum and Difference Identities for Cosine

Cofunction Identities cos (90° – θ) = sin θ

sin (90° – θ) = cos θ

cot (90° – θ) = tan θ

tan (90° – θ) = cot θ

csc (90° – θ) = sec θ

sec (90° – θ) = csc θ

Note: The angles θ and 90° – θ can be negative and / or obtuse.

Example 1

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Ch 5 Notes F07 O’Brien

Example 2

Example 3

II.

Sum and Difference Identities for Cosine cos (A + B) = cos A cos B – sin A sin B

[Functions stay together, operator changes.]

cos (A – B) = cos A cos B + sin A sin B

[Functions stay together, operator changes.]

Example 4

III.

Applying the Sum and Difference Identities A.

Reducing cos (A – B) to a Function of a Single Variable

Example 5

B.

Finding cos (s + t) Given Information about s and t

Example 6

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C.

Ch 5 Notes F07 O’Brien

Verification of an Identity

Example 7

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Section 5.4 I.

II.

Sum and Difference Identities for Sine sin (A + B) = sin A cos B + cos A sin B

[Functions mix; sign stays.]

sin (A – B) = sin A cos B – cos A sin B

[Functions mix; sign stays.]

Sum and Difference Identities for Tangent

tan (A + B) = III.

Sum and Difference Identities for Sine and Tangent

tanA + tanB 1 − tanAtanB

tan (A − B) =

tanA − tanB 1 + tanAtanB

Applying the Sum and Difference Identities A.

Finding Exact Sine and Tangent Function Values

Example 1

Example 2

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Ch 5 Notes F07 O’Brien

Example 3

B.

Writing Functions as Expressions Involving Functions of θ

Example 4

Example 5

C.

Finding Function Values and the Quadrant of A + B

Example 6

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D.

Ch 5 Notes F07 O’Brien

Verifying an Identity Using Sum and Difference Identities

Example 7

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Section 5.5 I.

Double-Angle Identities

Double-Angle Identities

2 tan (A) 1 − tan 2 (A)

sin (2A) = 2 sin (A) ⋅ cos(A)

tan (2A) =

cos (2A) = cos 2 (A) − sin 2 (A)

cos (2A) = 2 cos 2 (A) − 1

A.

cos (2A) = 1 − 2 sin 2 (A)

Finding Function Values of θ Given Information about 2θ

Example 1

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B.

Ch 5 Notes F07 O’Brien

Finding Function Values of 2θ Given Information about θ

Example 2

C.

Using an Identity to Write an Expression as a Single Function Value or Number

Example 3

Example 4

D.

Verifying a Double-Angle Identity

Example 5

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E.

Ch 5 Notes F07 O’Brien

Deriving a Multiple-Angle Identity

Example 5

II.

Product-to-Sum Identities

cos A ⋅ cos B =

1 [cos (A + B) + cos (A − B)] 2

sin A ⋅ sin B =

1 [cos (A − B) − cos (A + B)] 2

sin A ⋅ cos B =

1 [sin (A + B) + sin (A − B)] 2

cos A ⋅ sin B =

1 [sin (A + B) − sin (A − B)] 2

Using a Product-to-Sum Identity

Example 6

III.

Sum-to-Product Identities

⎛ A+B⎞ ⎛ A−B⎞ sin A + sin B = 2 sin ⎜ ⎟ ⋅ cos⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

⎛A+B⎞ ⎛ A−B⎞ sin A − sin B = 2 cos⎜ ⎟ ⋅ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

⎛ A−B⎞ ⎛ A+B⎞ cos A + cos B = 2 cos⎜ ⎟ ⎟ ⋅ cos⎜ ⎝ 2 ⎠ ⎝ 2 ⎠

⎛A−B⎞ ⎛ A+ B⎞ cos A − cos B = −2 sin ⎜ ⎟ ⎟ ⋅ sin ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠

Using a Sum-to-Product Identity

Example 7

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Ch 5 Notes F07 O’Brien

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Section 5.6 I.

Half-Angle Identities

Half-Angle Identities cos

A 1 + cos A =± 2 2

sin

A 1 − cos A =± 2 2

tan

A sin A = 2 1 + cos A

tan

A 1 − cos A = 2 sin A

tan

A 1 − cos A =± 2 1 + cos A

In the first three half-angle identities, the sign is chosen based on the quadrant of II.

A . 2

Applying the Half-Angle Identities A.

Using a Half-Angle Identity to Find an Exact Value

Example 1

B.

Finding Function Values of

θ Given Information about θ 2

Example 2

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C.

Ch 5 Notes F07 O’Brien

Finding Function Values of θ Given Information about 2θ

Example 3

D.

Using an Identity to Write an Expression as a Single Trigonometric Function

Example 4

Example 5

E.

Verifying an Identity

Example 6

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