Help with Trignometry (1 Viewer)

[Manwithaplan]

Hey,

I was wondering if anyone could help me out with these two problems. They're from fitzpatrick ex 24 (b) qs 59 and 62.

59. Find tan x in terms of tan (θ) if tan θ =cos(θ +x)/cos(θ-x)
62. If tan(A) =k tan(B), show that (k-1)*sin (A+B)=(k+1)*sin(A-B)

thanx in advance

 

[clintmyster]

iight 59) @ = theta

tan @ = (cos@cosx - sin@sinx) / (cos@cosx + sin@sinx)

tan @(cos@cosx + sin@sinx) = (cos@cosx - sin@sinx)

dividing both sides by 1 / (cos@cosx)

tan@tanx + tan²@tanx = 1 - tan@ + tanx

tanx(tan@ + tan²) = -tan@

tanx = -tan@ / (tan@ + tan²@)

= -tan@ / (1 + tan²@)



If thats not right leme know and Il have another shot



for 62) Show (k-1)*sin (A+B)=(k+1)*sin(A-B)

tanA = ktanB
sinA / cosA = ksinB / cosB

sinA = ksinBcosA / cosB
sinB = sinAcosB / kcosA


RHS
= (k+1)*(sinAcosB - sinBcosA)
= ksinAcosB - ksinBcosA + sinAcosB - sinBcosA
= ksinAcosB - kcosAsinAcosB / kcosA + ksinBcosAcosB / cosB - sinBcosA [substituting for sinA and sinB]
= ksinAcosB - sinAcosB + ksinBcosA - sinBcosA
= (k-1)sinAcosB + (k-1)sinBcosA
= (k-1)(sinAcosB + sinBcosA)
= [k-1][sin(A+B)]
= LHS


daym that took a while but I got it =] hope that helps!

 

[Manwithaplan]

thanks for the reply... i think the second question's proof is correct but the answer to the first question is (1-tanθ)/tanθ(1+tanθ)

 

[Timothy.Siu]

let theta =A

tan A=(cos A cos x - sin A sin x)/(cos A cos x + sin A sin x)

dividing top and bottom by cosA cos x

tan A=(1-tan A tanx)/(1+tan A tan x)

move to one side,
tan A + tan² A tan x= 1- tan A tan x
tan² A tan x+tan A tan x=1-tan A
tan x(tan²A+ tan A)=1-tan A
tan x=(1-tan A)/(tan²A+ tan A)

 

[clintmyster]

thanks for the reply... i think the second question's proof is correct but the answer to the first question is (1-tanθ)/tanθ(1+tanθ)

my bad

thats what happens when you do 3u trig the morning of ur 2u locus/sequences test xD

 

[Manwithaplan]

Thanx for ur help guys, much appreciated.

 

[Timothy.Siu]

no worries, its nice seeing people doing fitzpatrick questions =)

 

[pyrodude1031]

See solutions. Below.