Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean...

Ch5.1A – Using Fundamental IdentitiesReciprocal Identities

Quotient Identities

Pythagorean Identities

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

coscot

sintan

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)

csc csc2

cot cot2

cos cos2

Ex1) find all six trigs.

Ex2) Simplify: sinx.cos2x – sinx

,0tan and 2

3sec uu

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

cscsin

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

Ch5.1B – More IdentitiesEx5) Factor:

a) sec2θ – 1 b) 4tan2θ + tanθ – 3

Ex6) Factor: csc2x – cotx – 3

Ex7) Simplify: sint + cott.cost

Ex8) Rewrite not as a fractionxsin1

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

as a trig function. (0 < θ < π/2)

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

Ch5.1C p414 20 – 62 even

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!

Ex1) Verify:

sinsec

Ex2) Verify:

2sec2sin1

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

Ex4) Verify: tanx + cotx = secx.cscx

HW#25) Verify:

Ch5.2A p421 1 – 10 all

tansec

)sin(1

Ch5.2A p421 1 – 10 all

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

costansec

Ex6) Verify:

Ex7) Verify: tan4x = tan2x.sec2x – tan2x

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

HW#46) Verify:

Ch5.2B p421 21 – 39 odd

Ch5.2C p422 40-48all

Ch5.3A – Solving Trig Functions

Ex1) Solve: 2sinx – 1 = 0

Ex2) Solve: xx sin2sin

Ex3) Solve: 3tan2x – 1 = 0

Ex4) Solve: cotx.cos2x = 2cotx

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

Ex6) Solve: 2sin2x + 3cosx – 3 = 0

Ch5.3A p431 11 – 29odd,60

Ch5.3B – Solving Trig Functions cont

HW#18) Solve: tan23x = 3

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

#35) Solve:

#37) Solve:

Ch5.3B p432 12 – 22even, 31 – 37odd

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(

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(

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

tantan1

tantan)tan(

tantan1

tantan)tan(

Ex4) Prove the cofunction identity

tantan1

tantan)tan(

tantan1

tantan)tan(

xx sin2

Ex5) Simplify tan(θ + 3π)

tantan1

tantan)tan(

tantan1

tantan)tan(

Ex6) Solve

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

Ch5.4B – Sum and Difference Formulas cont

HW#12) Solve: tan255˚.

tantan1

tantan)tan(

tantan1

tantan)tan(

HW#14) Solve:

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

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)

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

tan2)2tan(

or = 2cos2u – 1 or = 1 – 2sin2u

Ex2) Find sin2θ, cos2θ, and tan2θ,

tan2)2tan(

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

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

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) =

Ch5.5B – Multiple Angle Formulas cont

Power Reducing Formulas:

Half Angle Formulas

HW#29) Simplify: sin2x.cos2x

2cos1 tan

2cos1cos

2cos1sin 222

Half Angle Formulas

HW#28) Simplify: sin4x

2cos1 tan

2cos1cos

2cos1sin 222

Half Angle Formulas

HW#33) Simplify:

2cos1 tan

2cos1cos

2cos1sin 222

Half Angle Formulas

HW#37) Simplify:

2cos1 tan

2cos1cos

2cos1sin 222

Half Angle Formulas

HW#39) Simplify: sin105˚

2cos1 tan

2cos1cos

2cos1sin 222

Half Angle Formulas

HW#43) Simplify:

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

2cos1 tan

2cos1cos

2cos1sin 222

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

Ch5 Rev p455 1 – 49 eoo, + 31

Ch5.1A – Using Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean...

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