Chapter 5 Quizlet
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
