C3 - Trig question

Hi,

could someone please give me a detailed description and working out for this question as I dont seem to understand how you do them....Iv got the basics of expanding the formulas etc but im really quite confused...!!

(i)Express 4 cosθ − sinθ in the form Rcos(θ + α ), where R > 0 and 0◦ < α < 90◦. [3]

(ii) Hence solve the equation 4 cosθ − sinθ = 2, giving all solutions for which −180◦ < θ < 180◦.
[5]

WILL REP THE MOST CONCISE ANSWER!!!
thank you :smile:

Reply 1

tcdc09

ok

i) R is the square root of 4 squared + 2 squared
To find the angle you do cos/sin as in 4/1 then do inverse tan of this answer

ii) you use the previuos answer to work out the angles :smile:

Reply 2

unamed

13

First expand the RHS:
4cosx-sinx=Rcosxcosa - Rsinxsina

[Mod edit: Please don't post full solutions. Read the posting guide sticky at the top of the thread list.]

(ii)use your Rcos(x+a) formula from (i) and solve! :biggrin:

Reply 3

Superman_Jr

13

I didn't help which is a good reason to rep me. :biggrin:

Reply 4

gottastudy

SK-mar

Hi,

could someone please give me a detailed description and working out for this question as I dont seem to understand how you do them....Iv got the basics of expanding the formulas etc but im really quite confused...!!

(i)Express 4 cosθ − sinθ in the form Rcos(θ + α ), where R > 0 and 0◦ < α < 90◦. [3]

(ii) Hence solve the equation 4 cosθ − sinθ = 2, giving all solutions for which −180◦ < θ < 180◦.
[5]

WILL REP THE MOST CONCISE ANSWER!!!
thank you :smile:

i) Use the cos(A+B) rule and expand. Remember to write 'R' in front. A = θ and B = α

Next step. You must group the terms together where you will find Rcosα and Rcosθ and that will be equal to a number.

You know that Rcosα = x and Rcosθ = y (whatever number you get, in this case 4 & 1).

Then, you draw a right angled triangle, fill in the lengths, O and A and work out R by using pythag. Work out what tana = O/A and do inverse tan to find out what a is. Now you have R and a. Re-write.

ii) From part i, you have formed an equation in the form of Rcos(θ + α ) = 2 (2's given in question)

Divide by the value of R by bringing it across to the RHS, so you get: 2/R

Solve like normal trig.

Hope it helps.

Reply 5

unamed

13

*MJ*

It works so why do I care.

You guys go back and do sina / cosa to obtain tana and then do tan^-1 right?

I find it's easier, just stick the R back in and obtain the angle.

:yeah:

yeah. But, it's funner to make it tan! Honest. :smile:

I don't like dealing with R, since it's usually a surd, and I hate surds!

Reply 6

unamed

13

SK-mar

sweet thanks......just wandering tho, strictly speaking, in case i get another type, should it be Rsina = -1? as its 4cosa - Sina.....

and could u actually do part ii.) for me so I fully understand, i will then rep you!!! thankssss

it's not -1 because it's also -Rsinxsina on the RHS. If that made any sense.

Reply 7

gottastudy

*MJ*

It works so why do I care.

You guys go back and do sina / cosa to obtain tana and then do tan^-1 right?

I find it's easier, just stick the R back in and obtain the angle.

:yeah:

I draw a right angled triangle. Label cos and sin, work out R using pythag. Then work out tana by O/A. :yeah:

Reply 8

*MJ*

2

unamed

yeah. But, it's funner to make it tan! Honest. :smile:

I don't like dealing with R, since it's usually a surd, and I hate surds!

True mathematicians love surds. :love:

I even eat them for breakfast. :mmm:

Reply 9

unamed

13

*MJ*

True mathematicians love surds. :love:

I even eat them for breakfast. :mmm:

I'm not a true mathematician, then. I'm a physicist - and we like our calculus. :smile:

We do it in our sleep.

[by the way, i swear you were a girl the other day! :eek:

]

Reply 10

SK-mar

OP

1

*MJ*

True mathematicians love surds. :love:

I even eat them for breakfast. :mmm:

right that was verrrrry heplful I completely get it now thankyou so much.....not just you but everyone who posted....Im gonna rep MJ but wish I could rep you all, sorry to those who I havnt!! but thanks anyway!!! ......... btw in these question will it most likely always be in the form
show that xcosa - xsina = Rcos(x-a) ....basicaly do they only use COS and SINE in C3, on the left hand side??

Reply 11

*MJ*

2

SK-mar

right that was verrrrry heplful I completely get it now thankyou so much.....not just you but everyone who posted....Im gonna rep MJ but wish I could rep you all, sorry to those who I havnt!! but thanks anyway!!! ......... btw in these question will it most likely always be in the form
show that xcosa - xsina = Rcos(x-a) ....basicaly do they only use COS and SINE in C3 on the left hand side??