Best solutions to - "I have the following function:f(x)=3cos⁡(x)+sin⁡(x)I am trying to..."

ikramkeyslo4s

Answered question

2022-04-01

I have the following function:

f (x) = \sqrt{3} \cos (x) + \sin (x)

I am trying to find what the amplitude of this function is without graphing it. I understand the formula for a cosine equation is:

y = A \cos (B x - C)

Theodore Davila

Beginner2022-04-02Added 14 answers

Remember your trig identities.

A \cos (x - C) = A \cos (x) \cos (C) + A \sin (x) \sin (C)

So if

f (x) = \sqrt{3} \cos (x) + \sin (x) = A \cos (x - C)

, then

A \cos (C) = \sqrt{3}

and

A \sin (C) = 1

. If we square both sides and add them together, we get

{[A \cos (C)]}^{2} + {[A \sin (C)]}^{2} = A^{2} [{\cos (C)}^{2} + {\sin (C)}^{2}] = A^{2} = {(\sqrt{3})}^{2} + {(1)}^{2} = 4

So A=2
From this argument it should be clear that in the general case, the amplitude of

a \sin (x) + b \cos (x)

\sqrt{a^{2} + b^{2}}

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