A support beam, within an industrial building, is subjected to vibrations along its length; emanating from
two machines situated at opposite ends of the beam. The displacement caused by the vibrations can be
modelled by the equation 𝑓(𝑥) = 3𝑠𝑖𝑛 (2𝑥 +𝜋3). Use the compound angle formulae to expand f(x) as a sum
of sine and cosine
Just use the formula
sin(a+b) = sina cosb + cosa sinb
You know if I honestly knew... I would not come here for any help.
You are never to no good use, and always hint to something and never assist. But thanks for your time.
To expand the equation 𝑓(𝑥) = 3𝑠𝑖𝑛 (2𝑥 + 𝜋/3) as a sum of sine and cosine, we can make use of the compound angle formulas.
The compound angle formulas allow us to express trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of the individual angles.
The cosine of the sum of two angles is given by:
cos(α + β) = cos(α) cos(β) - sin(α) sin(β)
The sine of the sum of two angles is given by:
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
In our case, we have 𝑓(𝑥) = 3𝑠𝑖𝑛 (2𝑥 + 𝜋/3). We can rewrite this expression as:
𝑓(𝑥) = 3[sin(2𝑥) cos(𝜋/3) + cos(2𝑥) sin(𝜋/3)]
Using the compound angle formulas, we can further simplify this expression as:
𝑓(𝑥) = 3[(sin(2𝑥) cos(𝜋/3)) + (cos(2𝑥) sin(𝜋/3))]
= 3[(sin(2𝑥) cos(𝜋/3)) + (cos(2𝑥) sin(𝜋/3))]
= 3[ (sin(2𝑥) * 1/2) + (cos(2𝑥) * √3/2) ]
This gives us the expanded form of 𝑓(𝑥) as a sum of sine and cosine.