Chapter 5 Quizlet

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rewrite sec2x as 1 + tan2x and factor

(Tanx+3)(tanx+2) = 0 tan x = -3 and tan x = -2 the period of the tangent function is pi x = arctan (-3) + pi, arctan (-3) + 2pi, arctan (-2) + pi, arctan (-2) + 2pi

cosx

1/1/cosx =

cscx

1/sinx

1/sinx - 2 (cosx/sinx) = 0

1/sinx (1-2cosx)=0 1/sinx = 0 when sinx approach's infinity which never occurs 1-2cosx = 0 when cosx = 1/2. This occurs when x = pi/3 and 5pi/3 The period of cosine is 2pi. To find the solution add 2npi where n 0, 1, 2,

5.3: Solving trigonometric equations

2sinx-1=0 Write original equation 2sinx = 1 Add 1 to each side Sinx =1/2 Divide by each side by 2 Quadratic type:many trigonometric equations are of quadratic type ax2 + bc + c =0 Quadratic in SINX 2sin2x - sinx -1 =0 2(sinx)2 - sinx-1=0

csc2x - cscx cotx = 1/sin2x - (1/sinx)(cos/sinx)

= 1-cosx/sin2x

cos4x = 0 and cosx-1 = 0

X=pi/8 + npi/4 cosx - 1 = 0 when x = 0

cos(a+b) = cos a cos b - sina sinb

cos(45 +120) = cos165

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

cos(x+pi/4) = cosx cos pi/4 - sinx sinpi/4 cos(x-pi/4) = cosx cos pi/4 + sinxsinpi/4

csc2x = 1 + cot2x

cot2x cos2x = (csc2x-1)cos2x = csc2xcos2x - cos2x = cos2x/sin2x - cos2x = cot2x - cos2x

csc (pi/2 - x) = secx

csc2(pi/2 - x) - 1 = sec2x -1 = tan2x

tanx = sinx/cosx = 4/5 / 3/5 = 4/3

reciprocal identities cscx = 1/sinx = 1/4/5 = 5/4 secx = 1/cos x = 1/3/5 = 5/3 cotx = 1/tanx =1/4/3 = 3/4 tanx = 4/3, cscx = 5/4, secx = 5/3, cotx = 3/4

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

sin(130 +50) = sin130cos50 = cos130sin50 = sin(180) = 0

factor: sinx(sinx-2)=0

sinx=0 when x = 0 and pi sin x = 2 has no solutions because 2 is out of the range of the function x = 0, pi

cot(pi/2 - x) = 1/tan(pi/2 -x)

tan (u-v) = tanu tanv/1+ tanu tan v 1/tan(pi/2 -x) = 1/tanpi/2-tanx = 1/1/tanx = tanx

(tan 63 - tan112) / (1+tan63tan112)

tan(63-112) tan(-49) or tan(311)

tan2 (csc2 - 1)

tan2(cot2) = tan2 (1/tan2) = 1

3tan2x = 1

tan2x= 1/3 Tanx = + 1/square root of 3 when x = pi/6, 5pi/6. 7pi/6. 11pi/6 the period of tangent is pi, to find the general solution, add npi, where n = 0, 1, 2...... X= pi/6 + npi and 5pi/6 + npi

1+tan2x= sec2x

the square root of 1 + tan2x = secx

5.5: Multiple-Angle and Product to Sum Formulas

Double angle formulas: sin2u = 2 sinu cosu cos2u = cos2u = sin2u = 2cos2u-1 = 1-2sin2u

2cos^2x - cosx-1=0

Factor: (2cosx +1)(cosx-1) = 0 cosx = 1 when x is a multiple of 2pi X = 0, 2pi/3, 4pi/3

5.1 Using fundamental identities

Reciprocal identities: SinU = 1/cscu Cscu = 1/sinu Cosu = 1/secu Secu=1/cosu Tanu = 1/cotu Cotu= 1/tanu Quotient Identities: tanu = sinu/cosu cotu = cosu/sinu Pythagorean Identities: sin2u+cos2u = 1 1 + tan2u = sec2u 1+cot2u = csc2u

5.4 sum and difference formulas

Sin(u+v) = sinu cosv + cosu sinv sin (u-v) = sinu cosv = cosu sin v Cos(u+v) = cosu cosv - sinu sinv Cos (u-v) = cosu cosv + sinu sinv

cosx = sin(pi/2 = x) = 1/the square root of 2

tanx = sinx/cosx = -1 cscx =1/sinx = -square root of 2/1 = negative square root of 2 secx = 1/cosx = square root of 2 cotx = 1/tanx = 1/-1 = -1

tanx = -1 when x = 3pi/4 and 7pi/4

the period of tangent is pi, to find the general solution, add n(pi), where n= 0,1,2... X= 3pi/4 + n(pi) and 7pi/4 + n(pi)

5.2 Verifying trigonometric identities

—> work with one side of the equation at a time —> look for opportunities to factor an expression, add fractions, square a binomial or create a monomial denominator —> look for opportunities to use the fundamental identities —> always try something

tan (u-v) = tanu- tanv/1+tanutanv

= (-4/3) - (-24/7)/1+ (-4/3)(-24/7) 44/21 over 1+96/21 = 44/21 times 21/117 = 924/2457. 44/117

(sin2x - sin4x) cosx = sin2xcosx (1- sin2x) = sin2x cosx cos2x =

= cos3x sin2x

tan ( pi/2 - x)secx = (cotx)(secx)

= cosx/sinx times 1/cosx = 1/sinx = cscx

sec2x - 1/secx -1

=(secx - 1)(secx + 1)/secx -1 = secx + 1


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