1. ## [SOLVED] Sine/Cosine Addition Formulae

Express x = 2sin(0.5t + 3) + 3cos(0.5t + 1) in the form x = Rsin(0.5t + ø)

All working must be done in radians.

I am stuck trying to solve this question.

I understand that:

asinx + bcosx can be written as Rsin(x + ø)

Where: Tanø = b/a and: R = sqrt(a² + b²)

The values for x (the frequency) in the question however have different numbers added to them. Does anyone know how to get around this? Thanks

2. Originally Posted by MathsDude69
Express x = 2sin(0.5t + 3) + 3cos(0.5t + 1) in the form x = Rsin(0.5t + ø)

All working must be done in radians.

I am stuck trying to solve this question.

I understand that:

asinx + bcosx can be written as Rsin(x + ø)

Where: Tanø = b/a and: R = sqrt(a² + b²)

The values for x (the frequency) in the question however have different numbers added to them. Does anyone know how to get around this? Thanks
Hi

You can expand

$2 \sin(0.5t+3) = 2 \cos 3 \sin 0.5t + 2 \sin 3 \cos 0.5t$

$3 \cos(0.5t+1) = 3 \cos 1 \cos 0.5t - 3 \sin 1 \sin 0.5t$

and sum

$2 \sin(0.5t + 3) + 3 \cos(0.5t + 1) = (2 \cos 3 - 3 \sin 1) \sin 0.5t + (2 \sin 3 + 3 \cos 1) \cos 0.5t$

and then use the way you have written

3. Hey:

So here's my attempt. Can someone please check whether ive gone completely wrong with this:

asinx + bcosx can be written as X = Rsin(x + ø)

Where: Tanø = b/a and: R = sqrt(a² + b²)

Expansion of terms =
2sin(0.5t + 3) = 2cos3sin0.5t + 2sin3cos0.5t
3cos(0.5t + 1) = 3cos1cos0.5t - 3sin1sin0.5t

2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t

Tanø = b/a

Tanø = 3/2

Tanø = 1.5

ø = Tan^-1(1.5)

ø = 0.983

R = sqrt(a^2 + b^2)

R = sqrt(4+9)

R = sqrt (13)

R = 3.606

Therefore x = 3.606sin(0.5t + 1.5)

Is this right or just completely wrong?? lol

2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t

Tanø = b/a

Tanø = (2sin3+3cos1) / (2cos3-3sin1)

R = sqrt(a^2 + b^2)

R = sqrt((2cos3-3sin1)²+(2sin3+3cos1)²)

R = sqrt (13 - 12 cos3 sin1 + 12 sin3 cos 1)

R = sqrt (13 + 12 sin2)