(x, y) (x, - y) (- x, - y) (- x, y). Sect 5.1 Verifying Trig identities ReciprocalCo-function...

(x, y)

(x, - y)(- x, - y)

(- x, y) cosx siny

cosx siny

cosx siny

cos

sintan

cos

sintan

tancos

sin

Sect 5.1 Verifying Trig identities

1sin

csc1

cossec

1tan

cot1

cottan

1sec

cos1

cscsin

Reciprocal

sin 90 cos

cos 90 sin

tan 90 cot

cot 90 tan

sec 90 csc

csc 90 sec

Co-function

sintan

coscos

cotsin

Quotient

2 2

2 2

2 2

sin cos 1

1 tan sec

1 cot csc

Pythagorean

sin sin

cos cos

tan tan

cot cot

sec sec

csc csc

Even/Odd

sec)(a

22

222

yxr

yxr

22 tan1sec

tan

1cot

3

5tan

3

5

cos

sintan

x

y

If and is in quadrant II, find each function value.S AT C

sin)(b cot)(cWhat Trig. Identity has tan and sec?

2

352 1sec

Negative answer.

9252 1sec

9342sec

334sec

Positive answer.

What Trig. Identity has tan and sin?

3453 22 r

34

345

34

5sin

r

y

Positive answer.

What Trig. Identity has tan and cot?

tan

1cot

tan

1cot

35

1cot

5

31

153

35

22 tan1sec

Write cos(x) in terms of tan(x). Secant has a relationship with both tangent and cosine.

1

tan1

cos

1 2

2

22

tan1

1cos

2tan1

1cos

2

2

tan1

tan1cos

Rationalize the denominator.

x2sec

xx 22 csc1cot

xx

2

2

csc1

cot1

Write in terms of sin(x) and cos(x), and simplify the expression so that no quotients appear.

xx 22 cot1csc x2cot x2csc x2csc x2cot

xx

2

2

csc1

cot1

xx

2

2

cot

csc

xxx

2

2

2

sincossin

1

xx

x 2

2

2 cos

sin

sin

1 x2cos

1

Sect 5.2 Verifying Trig identitiesGuidelines to follow.1. Work with one side of the equation at a time. It is often

better to work on the most complicated.

2. Look for opportunities to factor, add fractions, square binomials or multiply a binomial by it’s conjugate to create a monomial.

3. Look to use fundamental identities. Look to see what trig functions are in the answer.

4. Convert everything to sines and cosines

5. Always try something!


Verify.Distribute the cosecant.

Work on the right side first.

sincsccoscsc

Rewrite to sine and cosine.

Simplify the fractions.

Quotient Identity for cotangent.

1cot1cot

sincoscsc1cot

sinsin

1cos

sin

1

1

sin

cos


Verify.Work on the left side first.


Simplify the fractions by canceling .

Reciprocal Identity for secant.

xx 22 secsec

xxx 222 seccot1tan

xx

x22

2

sin

1

cos

sin

x2cos

1

Pythagorean Identity

1 + cot2x = csc2x xx 22 csctan


Verify.Work on the left side first.


Simplify the fractions by multiplying by the reciprocals and cancel.

Reciprocal Identity for secant and cosecant.

Rewrite the fraction as subtraction of two fractions with the same denominators.

22 cscsec

cossin

cottan

cossin

cot

cossin

tan

cossinsincos

cossincossin

cossin

1

sin

cos

cossin

1

cos

sin

22 sin

1

cos

1

2222 cscseccscsec


Verify.2

22

sec 1sin

sec


1 + tan2x = sec2x

tan2x = sec2x – 1

Work on the left side first.

2

2

sec

tan


2

2

2

cos1

cossin

Rewrite as multiplication.

1

cos

cos

sin 2

2

2

Cancel and Simplify.

22 sinsin


Verify. 2 1 12sec

1 sin 1 sin


sin2x + cos2x = 1

cos2x = 1 – sin2x

Reciprocal of cosine.

Work on the right side first. Two terms need to be condensed to one term. Find LCD and combine the fractions.

sin1

sin1

sin1

1

sin1

sin1

sin1

1

sin1sin1 LCD

2sin1

22 sin1

sin1

sin1

sin1

2sin1

2

22 cos

12

cos

2

22 sec2sec2


Verify. 2 2 2tan tan 1 cos 1 Work on the right side first. Pythagorean Identities.

sin2x + cos2x = 1

cos2x – 1 = – sin2x

1 + tan2x = sec2x 22 sinsec

Convert to cosine.

22

sincos

1

Multiply.

22 tantan

2

2

cos

sin


Verify. tan cot sec csc


sin2x + cos2x = 1

Work on the left side first. Try to combine the two terms into one.

sin

cos

cos

sin

Rewrite as two fractions multiplied together.

Reciprocals.

Convert to sine and cosine.

sincos LCD

cos

cos

sin

cos

sin

sin

cos

sin

sincos

cossin 22

sincos

1

sin

1

cos

1

cscseccscsec


Verify. cossec tan

1 sin

Work on the right side first. Two terms need to be condensed to one term.

cos

sin

cos

1 Convert to sine and cosine.

Combine.

cos

sin1When working with binomials, try multiplying by the conjugate to create differences of squares which will incorporate the Pythagorean Identities.

sin1

sin1

cos

sin1

sin1cos

sin1 2


sin2x + cos2x = 1

cos2x = 1 – sin2x

sin1cos

cos2

Cancel cosine.

sin1

cos

sin1

cos


Verify.2cot 1 sin

1 csc sin

Work on the left side first. Pythagorean Identity and convert to sine and cosine.

1 + cot2x = csc2x

cot2x = csc2x – 1

csc1

1csc2

csc2x – 1 is Diff. of Squares.

Factor.

csc1

1csc1csc

Cancel (csc x + 1)

1csc Convert to sine.

1sin

1

Combine.

sin

sin

sin

1

sin

sin1

sin

sin1

Sect 5.3 Sum and Difference Formulas

AA sin,cos

BB sin,cos

)0,1(

BABA sin,cos

212

212 yyxx

22 0sin1cos BABA

BABABA 22 sin1cos2cos

BA cos22

BABABA sinsin2coscos2cos2

BABABA sinsincoscoscos

A

BA – B

A – B Using Distance Formula

Dist. from (cos(A-B), sin(A-B)) to (1,0) Dist. from (cosA, sinA) to (cosB,sinB)

22 sinsincoscos BABA

BBAABBAA 2222 sinsinsin2sincoscoscos2cos

1 1cos2 BA 1 1 BAcoscos2 BAsinsin2BABA sinsin2coscos22

The Cosine of the Difference of Two Angles

F.O.I.L. F.O.I.L. F.O.I.L.

Pythagorean Identity Pythagorean IdentityPythagorean Identity

Subtract by 2.– 2 – 2

Divide by –2.2 2 2




The Cosine of the Difference of Two Angles

Substitute (-B) for B in the formula to make the Cosine of the Sum of Two Angle.

The Cosine of the Sum of Two Angles

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

90cossin

BABABA sin90sincos90cos90cos

BAsin

To make the Sine of the Sum & Difference of Two Angles we will need the Cofunction Identities for Sine and Cosine.

BA 90cos

90sincos

Start with . BA BABA 90cossin BA 90cos


BAsincosBAcossinSubstitute (-B) for B in the formula to make the Sine of the Sum of Two Angle.

BABABA sincoscossinsin

BABABA sincoscossinsin sin (– B) = – sin (B)cos (– B) = cos (B)

BA

BABA

cos

sintan

BABA

BABA

BABA

BABA

BA

coscossinsin

coscoscoscos

coscossincos

coscoscossin

tan

To make the Tangent of the Sum & Difference of Two Angles we will need the Quotient Identities for Tangent.

BABA

BABA

sinsincoscos

sincoscossin

Tricky manipulation: We want this fraction to have tangents in the formula. Need to divide by the same factor in both the top and bottom to make tangents.Start with where we need to divide by cosine.

cos (A)

cos (B)cos (A)

cos (B)

This is what we need divide by all the factors.

BA

BA

tantan1

tantan

BA

BABA

tantan1

tantantan

BABA

BABA

BA

BABA

sinsincoscos

sincoscossin

cos

sintan

BABA

BABA

BABA

BABA

coscossinsin

coscoscoscos

coscossincos

coscoscossin

BA

BA

tantan1

tantan

BA

BABA

tantan1

tantantan

sin (B)sin (A)

This is what we need divide by all the factors.

Find the exact value of .

12

7cos

10515712

1807

12

7

105cosUse the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

4560cos


45sin60sin45cos60cos4560cos45

60

30

1

1

2

3

12

2

2

2

3

2

2

2

1

4

62

4

6

4

2


43

5cos

3006053

1805

3

5

Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

45

60

30

1

1

2

3

1

2

2

2

2

3

2

2

2

1

454


45sin300sin45cos300cos45300cos

4

62

4

6

4

2

Suppose that for a Q2 angle and for a Q1 angle .

Find the exact value of each of the following.A. B. C. D.

13

12sin

sinsincoscoscos

5

12 13

5

3

13

12

5

4

13

5

5

3sin

cos cos cos cos

13

5cos

3

4

5

5

4cos

sinsincoscoscos

65

56

65

36

65

20

5

3

13

12

5

4

13

5

65

16

65

36

65

20

Find the exact value of . 75sin

75sinUse the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

4530sin


45sin30cos45cos30sin4530sin45

60

30

1

1

2

3

12

2

2

2

3

2

2

2

1

4

62

4

6

4

2


12

7tan

10515712

1807

12

7


4560

30 1

1

2

3

12

13

11

13

1

BA

BABA

tantan1

tantantan

45tan150tan1

45tan150tan45150tan

3

11

13

1

3

1

3

33

3

3

1

13

31

10545150

3

133

31

31

31


12

7tan

10515712

1807

12

7


45

60

301

1

23

1

2

3 1

1 3 1

tan tantan

1 tan tan

A BA B

A B

tan 60 tan 45tan 60 45

1 tan 60 tan 45

1 3

1 3

60 45 105

Find the exact value of . 160sin40cos160cos40sin

120sin 16040sin


2

360

30

3

1

2

120

Sect 5.5 Dble Angle, Power Reducing, and Half Angle FormulasDouble Angle Formulas: Revise the Sum of Sin, Cos, & Tan Formulas


BA

BABA

tantan1

tantantan

AAAAAA sincoscossinsin


AAA cossin22sin =>

AAAAAA sinsincoscoscos AAA 22 sincos2cos AAA 22 sinsin12cos AA 2sin212cos

AAA 22 cos1cos2cos 1cos22cos 2 AA

AA

AAAA

tantan1

tantantan

A

AA

2tan1

tan22tan

Substitute A in for B.

Find given and . sin 2 ,cos 2 , tan 2 5cos

13

32

2

5

13 1212513 22 y

cossin22sin

13

12sin

169

120

13

5

13

122

5

12tan

22 sincos2cos 169

119

13

12

13

522

2tan1

tan22tan

119

120

512

1

512

2

2

Choose one of the double angle identities to find a value for sine or cosine.

1cos22cos

sin212cos

sincos2cos

2

2

22

Substitute in 4/5.

Subtract by 1.

Divide by -2.

542cos 18090 Find the values of the six trigonometric functions of if and .

2

2

sin215

4

sin212cos

2sin25

1

2sin10

1

sin10

1

Square root both sides, but the answer will be positive, since we are Q2.

10

10

10

1sin

10

3110 22

1

10

103

10

3cos

3

1tan

3cot

3

10sec

10csc

SOH-CAH-TOA

Verify.

Work on the left side first. Convert to sine and cosine with Quotient Identity.

Double angle identity. 2sin(x) cos(x) = sin(2x)

Rewrite the double angle formula.

2cos2x – 1 = cos(2x)

2cos2x = 1 + cos(2x)

Cancel

2cos12sincot

2sinsin

cos

cossin2sin

cos

2cos2

2cos12cos1

2cossincos 22 xxx 72cos7sin7cos 22

x14cos

AAA cossin22sin

22

1

152sin2

1

15cos15sin

4

1

2

1

2

130sin

2

1

Find an identity for cos3 2cos


cossin2sin1cos2cos2cos 2 ( )( )

3 2cos 2 2cos cos 2sin cos

3 2cos 2 2cos cos 2 1 cos cos

3 3cos 2 2cos cos 2cos 2cos

34cos 3cos

3cos3 4cos 3cos

Substitute Dble angle Identity.

Pythagorean Identity, rewrite with all cosines.

2sinsin2coscos2cos ( )( )

3 2cos 2 2cos cos 2cos 1 cos

Distribute

BABABA cossinsincoscos


Product to Sum & Sum to Product FormulasHow to create the Product to Sum Formulas. Add and subtract Sum and Difference formulas for Sine and Cosine.



BABABA coscoscoscos2

BABABA coscos2

1coscos

BABABA coscossinsin2

BABABA coscos2

1sinsin

BABABA sinsincoscossin




BABABA sinsincossin2

BABABA sinsin2

1cossin

BABABA sinsinsincos2

BABABA sinsin2

1sincos

BABABA coscoscoscos 21

BABABA coscossinsin 21

BABABA sinsincossin 21

BABABA sinsinsincos 21

Product to Sum Formulas

Sum to Product Formulas

2

cos2

cos2coscosBABA

BA


BABABA coscoscoscos2

22cos

22cos

2cos

2cos2

22

yxyxyxyxyxyx

yxBand

yxALet

xyyxyx

coscos2

cos2

cos2

The reason we choose these two fractions for A and B is because we need two values that add up to x and two values that subtract to be y.


BABABA coscossinsin 21


BABABA sinsinsincos 21

Product to Sum Formulas

Sum to Product Formulas

2

cos2

cos2coscosBABA

BA

2

sin2

sin2coscosBABA

BA

2

cos2

sin2sinsinBABA

BA

2

cos2

sin2sinsinBABA

BA

Rewrite as a sum or difference of two functions

Rewrite using sums to product identity.

sin 6 cos 2x x


xxxxxx 26sin26sin2cos6sin 21

xx 4sin8sin21

xx 3cos4cos

2

sin2

sin2coscosBABA

BA

2

34sin

2

34sin23cos4cos

xxxxxx

2sin

2

7sin2

xx

Half Angle Formulas

2cos 2 1 2sinA A 2cos 2 2cos 1A A

2

2

sintan

2 cos

2Let A

22sin 1 cos 2A A

2 1 cos 2sin

2

AA

1 cossin

2 2

2 1 cos 2cos

2

AA

2 1 cos 2cos

2

AA

2

cos1

2cos

1 cos2

1 cos2

tan2

1 cos2

1 cos2

tan2

cos1

cos1

2tan

The + symbol in each formula DOES NOT mean there are 2 answers, instead it indicates that you must determine the sign of the trigonometric functions based on which quadrant the half angle falls in.

2Let A

cos1

cos1

2tan

cos1

cos1

2tan

cos1cos1

cos1cos1

2

2

2

2

sin

cos1

cos1

cos1

sin

cos1

2tan

cos1cos1

cos1cos1

2

2

2

2

cos1

sin

cos1

cos1

cos1

sin

2tan

Find the exact value for . 5.112cos

Verify the identity.

2sin

2cos1tan

2

225cos5.112cos

2

cos1

2cos

2

225cos1

2

225cos

2

22

1

S A

T C 2

2

222

22

2

22

4

22

cossin2

sin211 2

cossin2

sin2 2

cos

sintan

(x, y) (x, - y) (- x, - y) (- x, y). Sect 5.1 Verifying Trig identities ReciprocalCo-function...

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Transcript of (x, y) (x, - y) (- x, - y) (- x, y). Sect 5.1 Verifying Trig identities ReciprocalCo-function...