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Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u u u u u u u cot 1 tan sec 1 cos csc 1 sin sin cos cot cos sin tan u u u u u u

Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

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Page 1: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1A – Using Fundamental IdentitiesReciprocal Identities

Quotient Identities

Pythagorean Identities

sin2u + cos2u = 11 + tan2u = sec2u1 + cot2u = csc2u

uu

uu

uu

cot

1tan

sec

1cos

csc

1sin

sin

coscot

cos

sintan

u

uu

u

uu

Page 2: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Cofunction Identities

Even/Odd IdentitiesEven Odd

cos(–u) = cos(u) sin(–u) = –cos(u)sec(–u) = sec(u) tan(–u) = –tan(u)

cot(–u) = –cot(u)csc(–u) = –csc(u)

uuuu

uuuu

uuuu

sec2

csc csc2

sec

tan2

cot cot2

tan

sin2

cos cos2

sin

Page 3: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex1) find all six trigs.

Ex2) Simplify: sinx.cos2x – sinx

,0tan and 2

3sec uu

Page 4: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex3) Use ur calc to determine if the following are identities:

a) cos3x = 4cos3x – 3cosx

b) cos3x = sin(3x – )

Ex4) Verify by hand:

Ch5.1A p414 19 – 43odd

2

cscsin

cos

cos1

sin

Page 5: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1A p414 19 – 43odd Do 19,21,29,31,37,39,41 in class

Page 6: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1A p414 19 – 43odd Do 19,21,29,31,37,39,41 in class

Page 7: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1A p414 19 – 43odd Do 19,21,29,31,37,39,41 in class

Page 8: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1A p414 19 – 43odd Do 19,21,29,31,37,39,41 in class

Page 9: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B – More IdentitiesEx5) Factor:

a) sec2θ – 1 b) 4tan2θ + tanθ – 3

Ex6) Factor: csc2x – cotx – 3

Page 10: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex7) Simplify: sint + cott.cost

Ex8) Rewrite not as a fractionxsin1

1

Page 11: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex9) If x = 2tanθ, use substitution to express

as a trig function. (0 < θ < π/2)

Ch5.1B p414 45-63odd,71-75odd

24 x

Page 12: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B p414 45-63odd,71-75odd

Page 13: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B p414 45-63odd,71-75odd

Page 14: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B p414 45-63odd,71-75odd

Page 15: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B p414 45-63odd,71-75odd

Page 16: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1B p414 45-63odd,71-75odd

Page 17: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 18: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 19: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 20: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 21: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 22: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 23: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 24: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.1C p414 20 – 62 even

Page 25: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2A – Verifying Trig Identities

Guidelines: 1. Work one side at a time, usually the most complicated 1st. 2. Look to: - Factor

- Add fractions

- Square a binomial

- Get a monomial denominator 3. Use fundamental identities 4. Head toward sine and cosine

5. But try SOMETHING!

Page 26: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex1) Verify:

22

2

sinsec

1sec

Page 27: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex2) Verify:

2sec2sin1

1

sin1

1

Page 28: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex3) Verify: (tan2x + 1)(cos2x – 1) = –tan2x

Page 29: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex4) Verify: tanx + cotx = secx.cscx

Page 30: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

HW#25) Verify:

Ch5.2A p421 1 – 10 all

tansec

)sin(1

)cos(

Page 31: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2A p421 1 – 10 all

Page 32: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2A p421 1 – 10 all

Page 33: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2B – More Verifying Trig ID’sEx5) Verify:

y

yyy

sin1

costansec

Page 34: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex6) Verify:

sin

sin1

csc1

cot2

Page 35: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex7) Verify: tan4x = tan2x.sec2x – tan2x

Verify: sin3x.cos4x = (cos4x – cos6x).sinx

Page 36: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

HW#46) Verify:

Ch5.2B p421 21 – 39 odd

cot

1csc

1csc

cot

Page 37: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2B p421 21 – 39 odd

Page 38: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2B p421 21 – 39 odd

Page 39: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2B p421 21 – 39 odd

Page 40: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2B p421 21 – 39 odd

Page 41: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.2C p422 40-48all

Page 42: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3A – Solving Trig Functions

Ex1) Solve: 2sinx – 1 = 0

Ex2) Solve: xx sin2sin

Page 43: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex3) Solve: 3tan2x – 1 = 0

Ex4) Solve: cotx.cos2x = 2cotx

Page 44: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex5) Solve: 2sin2x – sinx – 1 = 0 over [0,2π]

Ex6) Solve: 2sin2x + 3cosx – 3 = 0

Ch5.3A p431 11 – 29odd,60

Page 45: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3A p431 11 – 29odd,60

Page 46: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3A p431 11 – 29odd,60

Page 47: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +
Page 48: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3B – Solving Trig Functions cont

HW#18) Solve: tan23x = 3

#24) Solve: cos2x.(2cosx + 1) = 0

Page 49: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

#35) Solve:

#37) Solve:

Ch5.3B p432 12 – 22even, 31 – 37odd

2

2

2cos

x

0cos1

cos1

x

x

Page 50: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3B p432 12 – 22even, 31 – 37odd

Page 51: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.3B p432 12 – 22even, 31 – 37odd

Page 52: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4A – Sum and Difference Formulas

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv

cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

Ex1) Find the exact value of cos75˚.

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

Page 53: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

Ex2) Find the exact value of , given that

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

12sin

4312

Page 54: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

Ex3) Find the exact value of sin42˚.cos12˚ – cos42˚.sin12˚

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

Page 55: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

Ex4) Prove the cofunction identity

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

xx sin2

cos

Page 56: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

Ex5) Simplify tan(θ + 3π)

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

Page 57: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ex6) Solve

Ch5.4A p440 7 – 25odd (7-sin,9-cos,11-sin,13-cos,15-tan)

14

sin4

sin

xx

Page 58: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4A p440 7 – 25odd (7-sin,9-cos,11-sin,13-cos,15-tan)

Page 59: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4A p440 7 – 25odd (7-sin,9-cos,11-sin,13-cos,15-tan)

Page 60: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4A p440 7 – 25odd (7-sin,9-cos,11-sin,13-cos,15-tan)

Page 61: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4B – Sum and Difference Formulas cont

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

HW#12) Solve: tan255˚.

tantan1

tantan)tan(

tantan1

tantan)tan(

vu

vuvu

vu

vuvu

Page 62: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

HW#14) Solve:

12

7sin

Page 63: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

HW#32) Prove: sin(x+π).sin(x–π) = sin2x

Page 64: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(u + v) = sinu.cosv + cosu.sinv sin(u – v) = sinu.cosv – cosu.sinv cos(u + v) = cosu.cosv – sinu.sinv cos(u – v) = cosu.cosv + sinu.sinv

HW#44) Prove: cos(x+y) + cos(x–y) = 2cosx.cosy

Ch5.4B p440 8 – 26 even, 32,44 (8-sin,10-cos,12-tan,14-sin,16-cos)

Page 65: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4B p440 8 – 26 even, 32,44 (8-sin,10-cos,12-tan,14-sin,16-cos)

Page 66: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4B p440 8 – 26 even, 32,44 (8-sin,10-cos,12-tan,14-sin,16-cos)

Page 67: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5A – Multiple Angle Formulas

sin(2u) = 2sinu.cosu cos(2u) = cos2u – sin2u

or = 2cos2u – 1 or = 1 – 2sin2u

Ex1) Find all the solutions of: 2cosx + sin2x = 0

tan1

tan2)2tan(

2 u

uu

Page 68: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(2u) = 2sinu.cosu cos(2u) = cos2u – sin2u

or = 2cos2u – 1 or = 1 – 2sin2u

Ex2) Find sin2θ, cos2θ, and tan2θ,

given

tan1

tan2)2tan(

2 u

uu

22

3for

13

5cos

Page 69: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

sin(2u) = 2sinu.cosu cos(2u) = cos2u – sin2u

or = 2cos2u – 1 or = 1 – 2sin2u

Ex3) Express sin3x in terms of sinx.

Ch5.5A p451 1 – 25odd, not 15Quiz tomorrow on Sum/Diff/Double Angle Formulas

Page 70: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5A p451 1 – 25odd, not 15Quiz tomorrow on Sum/Diff/Double Angle Formulas

Page 71: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5A p451 1 – 25odd, not 15Quiz today on Sum/Diff/Double Angle Formulas

Page 72: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5A p451 1 – 25odd, not 15Quiz today on Sum/Diff/Double Angle Formulas

Page 73: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5A p451 1 – 25odd, not 15Quiz today on Sum/Diff/Double Angle Formulas

Page 74: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.4,5.5 Quiz Name____________

Form A Form B1. sin(u + v) = 1. cos2θ =2. cos(u + v) = 2. tan2θ =3. sin(u – v) = 3. sin2θ=4. cos(u – v) = 4. cos(u + v) =5. tan(u – v) = 5. cos(u – v) =6. sin2θ= 6. sin(u + v) =7. tan2θ = 7. sin(u – v) =8. cos2θ = 8. tan(u – v) =

Page 75: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5B – Multiple Angle Formulas cont

Power Reducing Formulas:

Half Angle Formulas

HW#29) Simplify: sin2x.cos2x

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

Page 76: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Power Reducing Formulas:

Half Angle Formulas

HW#28) Simplify: sin4x

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

Page 77: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Power Reducing Formulas:

Half Angle Formulas

HW#33) Simplify:

5θ

12

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

2cos

Page 78: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Power Reducing Formulas:

Half Angle Formulas

HW#37) Simplify:

5θ

12

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

2csc

Page 79: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Power Reducing Formulas:

Half Angle Formulas

HW#39) Simplify: sin105˚

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

Page 80: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Power Reducing Formulas:

Half Angle Formulas

HW#43) Simplify:

Ch5.5B p452 27,28,29,33,34,35,37,39,40,41,43,44,45

u

uu

uu

uu

2cos1

2cos1 tan

2

2cos1cos

2

2cos1sin 222

cos1

sinor

sin

cos1

2tan

2

cos1

2cos

2

cos1

2sin

u

u

u

uu

uuuu

8csc

Page 81: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5B p452 27,28,29,33,34,35,37,39,40,41,43,44,45

Page 82: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5.5B p452 27,28,29,33,34,35,37,39,40,41,43,44,45

Page 83: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +
Page 84: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 85: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 86: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 87: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 88: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 89: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 90: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31

Page 91: Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 +

Ch5 Rev p455 1 – 49 eoo, + 31


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