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

• lionely
In summary, the conversation discusses the solution of an equation involving cosx and sinx for values of x from 0 to 360, and the question of which formula (cos(A-B) or sin(A+B)) is more advantageous in solving the equation. The solution using cos(A-B) is found to be easier and more accurate, while the solution using sin(A+B) is more complicated and may not satisfy the equation. The advantage of using cos(A-B) is that it involves fewer steps.
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?"

## The Attempt at a Solution

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

lionely said:

## 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?"

## 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?

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

lionely said:
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?

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.

lionely said:
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° .

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

## 1. What is the difference between cos(A-B) and sin(A+B)?

Cos(A-B) and sin(A+B) are both trigonometric functions that involve the sum and difference of angles. The main difference between the two is that cos(A-B) involves the difference of two angles, while sin(A+B) involves the sum of two angles.

## 2. How do I calculate the value of cos(A-B) and sin(A+B)?

To calculate the value of cos(A-B) or sin(A+B), you will need to know the values of A and B and the values of cos(A) and sin(B). You can then use the trigonometric identities of cos(A-B) and sin(A+B) to solve for the value.

## 3. What is the geometric interpretation of cos(A-B) and sin(A+B)?

The geometric interpretation of cos(A-B) and sin(A+B) involves the unit circle. The value of cos(A-B) represents the horizontal distance between the point (cos(A), sin(A)) and (cos(B), sin(B)), while the value of sin(A+B) represents the vertical distance between the same two points.

## 4. How does the value of a and b affect the function acos(x) + bsin(x)?

The values of a and b will affect the amplitude, period, and phase shift of the function acos(x) + bsin(x). The amplitude will be equal to the square root of (a^2 + b^2), the period will be 2π/√(a^2 + b^2), and the phase shift will be equal to arctan(b/a).

## 5. In what real-life situations would you use cos(A-B) and sin(A+B)?

Cos(A-B) and sin(A+B) are commonly used in physics and engineering to describe the relationship between two waves or oscillating systems. They can also be used in navigation, such as determining the position of a ship based on the observed angles to two known landmarks.

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