# Quick Trig Question

#### 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

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

MHF Hall of Honor
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
$$\displaystyle \cos(P+Q) = \cos P \cos Q - \sin P \sin Q$$
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):
$$\displaystyle \tan C = -\tfrac45$$