trigonometry chapter 5 lecture notes section 5.1 ...faculty.nwacc.edu/tobrien/trig resources...

LHS Trig 8th ed Ch 5 Notes F07 O’Brien

1

Trigonometry Chapter 5 Lecture Notes

Section 5.1 Fundamental Identities

I. 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

θsin

1θ csc = θcos

1θ sec = θtan

1θcot =

III. Quotient Identities

θ cosθsin θtan =

θsin θ cosθcot =

IV. Pythagorean Identities

1θcosθsin 22 =+ θsec1θtan 22 =+ θcscθcot1 22 =+

V. Using the Fundamental Identities

A. Finding Trigonometric Function Values Given One Value and the Quadrant

Example 2 Given 31 cot x −= and x is in quadrant IV, find sin x. (modified #6)

xcot1 x csc x cot1 xcsc 222 +±=→+= ; cosecant and sine are negative in IV;

310

910

311 x csc

2−=−=⎟

⎠⎞

⎜⎝⎛−+−= →

10103

103

xcsc1sin x −=−==

Now find the three remaining trigonometric functions of x.

3cot x

1tan x −==

sin xcot x xcos ⋅= → 1010

10103

31 xcos =−⋅−=

10 xcos

1 xsec ==


2

B. Using Identities to Rewrite Functions and Expressions

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

sin x

xcoscot x = and from 1xcosxsin 22 =+ we know xsin1 xcos 2−±=

therefore, sin x

xsin1cot x2−±

= .

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

1θsin θcosθsin θ cosθsin

θsin θ cos

θcos1θsin θcot θ sec =

⋅⋅

=⋅⋅=⋅⋅

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 Verifying Trigonometric Identities

I. 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. II. 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 xcos

sin xtan x = is equivalent to tan x xcossin 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 1sin3sin2 2 ++ x x can be factored to ( )( )1sin1sin2 ++ xx (#17)


3

Example 4 xx cos

1sin

1+ can be added by getting a common denominator

xxxx

xx

xxx

xxx cossinsincos

sinsin

cos1

coscos

sin1

cos1

sin1

⋅+

=⋅+⋅=+

D. Sometimes it is helpful to express all trigonometric functions on one side of an identity in terms of sine and cosine.

Example 5 Verify xxx sin

sectan

= (#34)

xxxx

x

xx

xx sin

1cos

cossin

cos1

cossin

sectan

=⋅==

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 xxx

x cotcscsin

cos1+=

+

xxxx

xxx cotcsc

sincos

sin1

sincos1

+=+=+

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.

Example 7 Verify xxxx cossec

cossin 2

−= (#42)

xxxx

xxx

xx cossec

coscos

cos1

coscos1

cossin 222

−=−=−

=

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 xx

x 2cossin1

sin11 −

=+

.

xx

xx

xx

xx 22 cossin1

sin1sin1

sin1sin1

sin11

sin11 −

=−

−=

−−

⋅+

=+


4

H. 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.

Example 9 Verify ( )x

xx 222

sin11cot1tan

−=+ .

( )xx

xxx

xxx 2222

2222

sin11

cos1sec1tan

tan11tancot1tan

−===+=⎟

⎠

⎞⎜⎝

⎛ +=+

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.”

Example 10 ( )xxxx

sin1sin1tansec 2

+−

=−

working on left side: working on right side:________

( ) =− 2tansec xx =+−

xx

sin1sin1

=+− xxxx 22 tantansec2sec =−−

⋅+−

xx

xx

sin1sin1

sin1sin1

=+⋅−xx

xx

xx 2

2

2 cossin

cossin

cos12

cos1 =

−+−

xxx

2

2

sin1sinsin21

xx

xx

x 2

2

22 cossin

cossin2

cos1

+− =+−

xxx

2

2

cossinsin21

xx

xx

x 2

2

22 cossin

cossin2

cos1

+−

************************************************************************************ Section 5.3 Sum and Difference Identities for Cosine

I. 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


5

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


6

C. Verification of an Identity

Example 7

************************************************************************************ Section 5.4 Sum and Difference Identities for Sine and Tangent

I. 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.]

II. Sum and Difference Identities for Tangent

( )tanAtanB1

tanBtanABAtan−

+=+ ( )

tanAtanB1tanBtanABAtan

+−

=−

III. Applying the Sum and Difference Identities

A. Finding Exact Sine and Tangent Function Values

Example 1

Example 2


7

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


8

D. Verifying an Identity Using Sum and Difference Identities

Example 7

************************************************************************************ Section 5.5 Double-Angle Identities

I. Double-Angle Identities

cos(A)(A)sin 2(2A)sin ⋅= (A)tan1

(A) tan 2(2A)tan 2−=

(A)sin(A)cos(2A) cos 22 −= 1(A)cos 2(2A) cos 2 −= (A)sin 21(2A) cos 2−=

A. Finding Function Values of θ Given Information about 2θ

Example 1


9

B. 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


10

E. Deriving a Multiple-Angle Identity

Example 5

II. Product-to-Sum Identities

( ) ( )[ ]BA cosBA cos21B cosA cos −++=⋅ ( ) ( )[ ]BA cosBA cos

21Bsin Asin +−−=⋅

( ) ( )[ ]BAsin BAsin 21B cosAsin −++=⋅ ( ) ( )[ ]BAsin BAsin

21Bsin A cos −−+=⋅

Using a Product-to-Sum Identity

Example 6

III. Sum-to-Product Identities

⎟⎠⎞

⎜⎝⎛ −

⋅⎟⎠⎞

⎜⎝⎛ +

=+2

BAcos2

BAsin 2Bsin Asin ⎟⎠⎞

⎜⎝⎛ −

⋅⎟⎠⎞

⎜⎝⎛ +

=−2

BAsin2

BAcos 2Bsin Asin

⎟⎠⎞

⎜⎝⎛ −

⋅⎟⎠⎞

⎜⎝⎛ +

=+2

BAcos2

BAcos 2B cosA cos ⎟⎠⎞

⎜⎝⎛ −

⋅⎟⎠⎞

⎜⎝⎛ +

−=−2

BAsin2

BAsin 2B cosA cos

Using a Sum-to-Product Identity

Example 7


11

************************************************************************************ Section 5.6 Half-Angle Identities

I. Half-Angle Identities

2

A cos12Acos +

±= 2

A cos12Asin −

±= A cos1A cos1

2Atan

+−

±=

A cos1

Asin 2Atan

+=

Asin A cos1

2Atan −=

In the first three half-angle identities, the sign is chosen based on the quadrant of 2A .

II. Applying the Half-Angle Identities

A. Using a Half-Angle Identity to Find an Exact Value

Example 1

B. Finding Function Values of 2θ Given Information about θ

Example 2


12

C. 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

trigonometry chapter 5 lecture notes section 5.1 ...faculty.nwacc.edu/tobrien/trig resources...

Documents