# Quick Trig Question

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• June 4th 2010, 05:52 AM
Paymemoney
Quick Trig Question
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
• June 4th 2010, 05:54 AM
mathaddict
Quote:

Originally Posted by 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

in general ,

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

where $R=\sqrt{a^2+b^2}$

and $\tan \alpha=\frac{b}{a}$

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

but you flip the sign
• June 4th 2010, 07:31 AM
Grandad
Hello Paymemoney
Quote:

Originally Posted by 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
$\cos(P+Q) = \cos P \cos Q - \sin P \sin Q$
as follows:
$A\cos(2x+C) = 4\sin 2x +5\cos 2x$

$\Rightarrow A\cos2x\cos C -A\sin2x\sin C = 4\sin2x+5\cos2x$

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

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