Quick Trig Question

Dec 2008
509
2
Hi

Can someone tell me how to do this question:

1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

P.S
 
Sep 2008
1,261
539
West Malaysia
Hi

Can someone tell me how to do this question:

1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

P.S
in general ,

\(\displaystyle a\sin \theta+b\cos \theta=R\sin(\theta+\alpha)\)

where \(\displaystyle R=\sqrt{a^2+b^2}\)

and \(\displaystyle \tan \alpha=\frac{b}{a}\)

It works the same for \(\displaystyle a\cos \theta+b\sin \theta=R\cos (\theta-\alpha)\)

but you flip the sign
 

Grandad

MHF Hall of Honor
Dec 2008
2,570
1,416
South Coast of England
Hello Paymemoney
Hi

Can someone tell me how to do this question:

1) Convert 4sin2x+5cos2x into a single cosine wave form Acos(2x+C)

P.S
Just to expand a little on mathaddict's reply, use the identity
\(\displaystyle \cos(P+Q) = \cos P \cos Q - \sin P \sin Q\)
as follows:
\(\displaystyle A\cos(2x+C) = 4\sin 2x +5\cos 2x\)

\(\displaystyle \Rightarrow A\cos2x\cos C -A\sin2x\sin C = 4\sin2x+5\cos2x\)

\(\displaystyle \Rightarrow \left\{\begin{array}{l l}A\cos C = 5 & \quad\text{(1)}\\ A\sin C = -4&\quad\text{(2)}\end{array}\right.\)
Square (1) and (2) and add:
\(\displaystyle A^2 = 5^2+(-4)^2\)

\(\displaystyle \Rightarrow A = \sqrt{41}\)
Divide (2) by (1):
\(\displaystyle \tan C = -\tfrac45\)
Grandad