How do you express sin4x as a trigonometric funcition of x using the identities? I think that you are supposed to start with the sine sum identity and then use the double angle identities for sin2A and cos2A, but after that I get stuck...

Well, trying to express sin(4x) using trigonometric identities can be a bit tricky, but let's give it a shot!

We can start by using the double angle identity for sine, which states that sin(2A) = 2sin(A)cos(A).

So, let's take sin(4x) and rewrite it as sin(2(2x)). Applying the double angle identity, we have:

sin(4x) = 2sin(2x)cos(2x).

Now, we can express sin(2x) and cos(2x) in terms of sine and cosine functions of x using the double angle identities again.

sin(2x) = 2sin(x)cos(x).

cos(2x) = cos^2(x) - sin^2(x), where cos^2(x) means (cos(x))^2.

Substituting these expressions into our previous equation, we get:

sin(4x) = 2(2sin(x)cos(x))(cos^2(x) - sin^2(x)).

Now, we need to simplify this further by using the identity cos^2(x) - sin^2(x) = cos(2x). Let's substitute that in:

sin(4x) = 2(2sin(x)cos(x))(cos(2x)).

Now, we can simplify the expression further by multiplying:

sin(4x) = 4sin(x)cos(x)cos(2x).

So, there you have it! sin(4x) can be expressed as 4sin(x)cos(x)cos(2x).

Remember, math may not always be the funniest thing, but it certainly has its own twisted sense of humor!

To express sin(4x) as a trigonometric function of x using the identities, you are on the right track. Here's how you can proceed step-by-step:

1. Start with the sine sum identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).

2. Let A = 3x and B = x; therefore, sin(4x) = sin(3x + x).

3. Apply the sine sum identity to sin(3x + x): sin(3x + x) = sin(3x)cos(x) + cos(3x)sin(x).

4. Rewrite cos(3x) using the double angle identity: cos(3x) = cos(2x + x) = cos(2x)cos(x) - sin(2x)sin(x).

5. Substitute the values back into the expression from step 3: sin(4x) = sin(3x)cos(x) + cos(3x)sin(x)
= (sin(3x)cos(x)) + (cos(2x)cos(x) - sin(2x)sin(x))sin(x).

6. Use the double angle identities: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x).

7. Substitute the double angle identities into the expression from step 5: sin(4x) = (sin(3x)cos(x)) + ((cos^2(x) - sin^2(x))cos(x) - (2sin(x)cos(x))sin(x))sin(x).

8. Simplify the expression: sin(4x) = sin(3x)cos(x) + (cos^2(x)cos(x) - sin^2(x)cos(x) - 2sin^2(x)cos(x))sin(x).

9. Further simplify: sin(4x) = sin(3x)cos(x) + cos^3(x)sin(x) - sin^3(x)cos(x) - 2sin^3(x)cos(x).

10. Combine like terms: sin(4x) = sin(3x)cos(x) - 2sin^3(x)cos(x) + cos^3(x)sin(x) - sin^3(x)cos(x).

And there you have it! sin(4x) expressed as a trigonometric function of x using the identities.

To express sin(4x) as a trigonometric function of x, you're on the right track by using the identities for sine sum, double angle, and possibly even triple angle.

1. Start with the sine sum identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
In this case, let A = 3x and B = x:
sin(3x + x) = sin(3x)cos(x) + cos(3x)sin(x).

2. Use the double angle identities for sin(2A) and cos(2A):
sin(2A) = 2sin(A)cos(A) and cos(2A) = cos^2(A) - sin^2(A).

3. Express sin(3x) in terms of sin(x) using the double angle identity:
Since 3x = 2x + x:
sin(3x) = sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x).

4. Replace sin(3x) in the initial equation with the expression from step 3:
sin(3x + x) = (sin(2x)cos(x) + cos(2x)sin(x))cos(x) + cos(3x)sin(x).

5. Use the double angle identity for cos(2x) and expand the equation:
cos(2x) = cos^2(x) - sin^2(x).
sin(3x + x) = sin(2x)cos(x) + (cos^2(x) - sin^2(x))sin(x) + cos(3x)sin(x).

6. Simplify the equation further:
sin(3x + x) = sin(2x)cos(x) + cos^3(x)sin(x) - sin^3(x)sin(x) + cos(3x)sin(x).

7. Use the triple angle identity for sin(3x):
sin(3x) = 3sin(x) - 4sin^3(x).
Replace sin(3x) in the equation from step 6 with 3sin(x) - 4sin^3(x):
sin(4x) = sin(2x)cos(x) + cos^3(x)sin(x) - sin^3(x)sin(x) + cos(3x)sin(x).

Now, you have expressed sin(4x) as a trigonometric function of x using the given identities.

sin 4x

= 2sin(2x)cos(2x)
= 2(2sinxcosx)(1 - 2sin^2 x)
= 4sinxcosx(1 - 2sin^2 x)

this is expressed in terms of trig functions of x.