# Converting Expression acos(t)+bsin(t) into the Form Asin(t+alpha)

Any expression of the form acos(t)+bsin(t) can be written in the form Asin(t+alpha).

To do this let's factor out sqrt(a^2+b^2):

acos(t)+bsin(t)=sqrt(a^2+b^2)(a/(sqrt(a^2+b^2)) cos(t)+b/(sqrt(a^2+b^2))sin(t)).

But (a/sqrt(a^2+b^2))^2+(b/sqrt(a^2+b^2))^2=(a^2)/(a^2+b^2)+(b^2)/(a^2+b^2)=(a^2+b^2)/(a^2+b^2)=1. Therefore, from identity sin^2(alpha)+cos^2(alpha)=1 it follows that exists such angle alpha that sin(alpha)=a/(sqrt(a^2+b^2)) and cos(alpha)=b/(sqrt(a^2+b^2)).

If we denote sqrt(a^2+b^2) through A then we can write that

color(red)(acos(t)+bsin(t)=sqrt(a^2+b^2)(a/(sqrt(a^2+b^2)) cos(t)+b/(sqrt(a^2+b^2))sin(t)))color(red)(=)

color(red)(=A(sin(alpha)cos(t)+cos(alpha)sin(t))=Asin(t+alpha)).

Numbers a,b,A,alpha are connected by following relations:

• a=Asin(alpha),
• b=Acos(alpha),
• A=sqrt(a^2+b^2),
• sin(alpha)=a/(sqrt(a^2+b^2))=a/A,
• cos(alpha)=b/sqrt(a^2+b^2)=b/A.

Example. Convert 3sin(2t)+4cos(2t) into the form Asin(2t+alpha).

First of all note that coefficient a is coefficient near cosine and coeffcient b is coeffcient near sine.

That's why a=4, b=3, so A=sqrt(a^2+b^2)=sqrt(4^2+3^2)=sqrt(25)=5, sin(alpha)=a/A=4/5, cos(alpha)=b/A=3/5.

Therefore, 3sin(2t)+4cos(2t)=5sin(2t+alpha) where sin(alpha)=4/5,cos(alpha)=3/5.