Cos(A-B) vs sin(A+B) [acos(x) + bsin(x) question]

[lionely]

Homework Statement

I already solved the question but there was a question at the end just for thought I guess.

Solve the equation 3 cosx + 4 sinx = 2, for values of X from 0 to 360, inclusive.

Again I already solved it ,the thing that I am curious about is the question below in bold.

" What advantage is there in using the formula for cos(A-B) , rather than that for sin(A+B) in the above question?"

Homework Equations

The Attempt at a Solution

I don't see any advantages. But I am not entirely sure?

 

[SammyS]

Homework Statement

I already solved the question but there was a question at the end just for thought I guess.

Solve the equation 3 cosx + 4 sinx = 2, for values of X from 0 to 360, inclusive.

Again I already solved it ,the thing that I am curious about is the question below in bold.

" What advantage is there in using the formula for cos(A-B) , rather than that for sin(A+B) in the above question?"

Homework Equations

The Attempt at a Solution

I don't see any advantages. But I am not entirely sure?

What steps did you follow when you solved it?

 

[lionely]

cosycosx + sinysinx = constant

Comparing this with
3cosx + 4sinx = 2

so
(cosy)/3 = (siny)/4 that is tany = 4/3

so y = 53.13
and siny = 4/5 and cos y = 3/5 therefore

3/5cosx + 4/5sinx = 2/5

therefore cosxcosy +sinxsiny = 0.4

cos(x-y) = 0.4
x- 53.13 = 66.42 or 293.58
so x = 119.6 and 346.7

 

[SammyS]

cosycosx + sinysinx = constant

Comparing this with
3cosx + 4sinx = 2

so
(cosy)/3 = (siny)/4 that is tany = 4/3

so y = 53.13
and siny = 4/5 and cos y = 3/5 therefore

3/5cosx + 4/5sinx = 2/5

therefore, cosxcosy +sinxsiny = 0.4

cos(x-y) = 0.4
x- 53.13 = 66.42 or 293.58
so x = 119.6 and 346.7

So, that's the solution using cos(A - B).

The solution using sin(A + B) must have sin(y)cos(x) + cos(y)sin(x) = constant .

Giving tan(y) = 3/4, so that y ≈ 36.87° and sin(x + y) = sin(x + 36.87°) =0.4

Thus x + 36.87° ≈ 23.58° or 156.42° , plus multiples of 360° for each .

So, why is the cos(x-y) method easier to work with in this case?

 

[lionely]

Hmm because the cos(x-y) gives the write answers? When I put the answers gotten from the sin(A+B) it doesn't satisfy the equation.

 

[SammyS]

Hmm because the cos(x-y) gives the write answers? When I put the answers gotten from the sin(A+B) it doesn't satisfy the equation.

You will get the right numbers eventuality.
x + 36.87° ≈ 23.58° gives -13.27° . Add 360° to that to get346.71° .

So, what's the advantage/disadvantage?

 

[lionely]

Oh , well all I can say is that the cos(A-B) has less steps.