Question

How would you transform f(x) = 3 sin(4x - π) + 4 into a cosine function in the form f(x) = a cos(bx - c) + d?

Answer 1

\[Sin(\frac{\pi}{2} - x) = Cos(x)\]

But we have to be careful, as you have a 4 in front of your x.

We'll also need: Sin(-x) = - Sin(x)

Answer 2

So what do we do with the 4?

Answer 3

Glad I checked on that, It actually doesn't cause us a problem :)

So first, let's rewrite sin(4x-pi) by factoring out a negative from the argument.

Answer 4

For example:
\[Sin(x-a) = Sin(-(a-x)) = - Sin(a-x)\]

We want it to look something like the form:
\[Sin(\frac{\pi}{2} - x)\]

Answer 5

Wait... but if I factor out a negative, I get f(x)= 3-sin(-4x + pi)...

Answer 6

Unless that's right...

Answer 7

We aren't done yet, but the 3 is multiplying the Sin, it looks like, so you'd get:

\[-3 Sin(\pi - 4x)\]

Answer 8

There is another way to do this, if you'd rather use the angle addition formulas.

Sin(a+b) = Sin(a)Cos(b)+Cos(a)Sin(b)

Answer 9

I don't care. I just need to know how to do it this one time.

Answer 10

@Vandreigan

Answer 11

Well, let's use that way. I think it'll be a little more clear.

Use that formula on your original Sin(4x-pi). a = 4x, b = -pi

Answer 12

Sin(4x-pi) = Sin(4x)Cos(-pi) + Cos(4x)Sin(-pi)

right?

Answer 13

Right!

Now, we know what Cos(-pi) and Sin(-pi) are, so just evaluate those.

Answer 14

Sin(4x)(-1) + Cos(4x)(0)

Answer 15

Yep, which equals what?

Answer 16

-sin(4x) + 0
-sin(4x)

Answer 17

But I have to make it a cosine function... how do I do that?

Answer 18

Right, sorry.

We know that Sin(x) = Cos(pi/2 - x)

Using this, we can make it a cosine.

Answer 19

-sin(4x) = cos(pi/2-4x)

Where would the negative sign go?

Answer 20

You keep the negative sign in front.

-sin(4x) = -cos(pi/2 - 4x)

Answer 21

Now, notice we didn't do anything to the +4 to the right. It just tags along, as well.

Answer 22

So the answer would be f(x)= -cos(pi/2-4x) +4

Answer 23

That's the one :)

There's another answer, as well:

-sin(4x)+4 = sin(-4x)+4 = cos(pi/2 + 4x) + 4

in case they don't want to take that one (both are correct)

Answer 24

Okay! THANKS SO MUCH!!!! (: