equivalent expressions

Question 1

find the maximum value of 2sinx+3cosx
and the value of x^o(0≤ x ≤ 360) at which the maximum occurs

Answer 1

rsin(x+a)= rsinxcosa +rcosxsina

rcosa=2, rsina=3

r²cos²a + r²sin²a= 2²+3²=13

r²(cos²a + sin²a)==13

r²(1)=13

r=+-sqrt(13)

must be positive value to fit limits +sqrt(13)

rsina/rcosa=3/2 tana=1.5 a=56^o

therefore 2sinx+3cosx=sqrt(13)sin(x+56)

when sin(x+56)=1

max value= sqrt(13)

sin(x+56)=1 when (x+56)=90

x=34

only one solution between 0 and 360

Question 2

a) express 4cosx+3sinx as Rcos(x-a)

R>0, 0

b) solve 4cosx+3sinx=2 for 0<x>
</x>

Answer 2

Rcos(x-a)= Rcosxcosa + Rsinxsina

Rcosa=4, Rsina=3

tana=3/4 a=0.64radians

r²(cos²a+sin²a)=4²+3²=16+9=25

r=5

4cosx+3sinx=2

5cos(x-0.64)=2

x=1.8

only solution between 0 and π

Question 3

a) express cosØ-sinØ in th form rcos(Ø+a)
b) solve cosØ-sinØ=0.5 for 0≤Ø≤π

Answer 3

a) rcos(Ø+a) =RcosØcosa-rsinØsina

Rcosa=1, Rsina=1

r²(sin²a +cos²)=1²+1²

r=sqrt(2)

tana=1 a=π/4

cosØ-sinØ= sqrt(s)cos(Ø+π/4)

b) cosØ-sinØ=0.5 —> sqrt(2)cos(Ø+π/4)=0.5

Ø=0.42, only solution between 0 and π/2

Question 4

9sinØ-6cosØ=7 for 0^o≤Ø≤360^o

Answer 4

Rsin(Ø-a)=RsinØcosa-RcosØsina

rcosa=9, Rsina=6

r²(cos²a+sin²a)=9²+6²=117

r=sqrt(117)

tana=6/9=1/3, a=33/7^o

sqrt(117)sin(Ø-33.7)=7

(Ø-33.7)=40.33^o and 139.67^o

Ø=74^o, 173.4^o

Question 5

asinx+bcosx can be written in the forms

Answer 5

rsin(x+a), where a =rcosa and b=rsina

rcos(x-a), where a=rsina and b=rcosa

Question 6

asinx-bcosx can be written in the form

Answer 6

rsin(x-a), where a=rcosa and b=rsina

rcos(x+a), where a=-rsina and b=-rcosa