Pre-calc Half Angles?

Question

How would I "Express in terms of a single angle (x)" for cos(3x) and sin(4x)?

So far for cos(3x), I have:

cos(3x) = cos(2x + x)
= cos 2x cos x - sin 2x sin x
= (2 cos ^2 x - 1)cos x - 2sin^x cos x
...and then I'm lost

As for sin(4x), I have down...
sin(4x) = sin (2x + 2x)
= sin 2x cos 2x + cos2x sin 2x
= 2 sin 2x cos 2x
= 2 (2 sinx cosx) (cos^2 x - sin^2 x)
...and that's all I have for that.

There's several formulas for cos (2x) and sin (2x), but I have no idea which to use. Any help would be great.

Thanks.

Answer 1

Looks like your main problem is that you couldn't recognize when you had finished the problem. You have already worked both problems to completion; all that is left is to simplify the result.

After simplifying your result, you get
cos (3x) = (2 cos² x - 1)cos x - 2 sin² x cos x
= 2 cos³ x - cos x - 2(1 - cos² x) cos x
= 4 cos³ x - 3 cos x

The sin(4x) one is similar. You already have
sin (4x) = 4 sin x cos x (cos² x - sin² x)
= 4 sin x cos x (1 - 2 sin² x)
= 4 cos x (sin x - 2 sin³ x)
except you have a cos x term. If you must convert this to √(1 - sin² x), you can.

Answer 2

There are multiple formulas that are mathematically equivalent. Here is what it sounds like you are looking for.

cos 3x = cos(2x + x) = (cos 2x)(cos x) - (sin 2x)(sin x)

= (2cos²x - 1)(cos x) - 2(sin x)(cos x)(sin x)

= (2cos³x - cos x) - 2(sin²x)(cos x)

= (2cos³x - cos x) - 2(1 - cos²x)(cos x)

= 2cos³x - cos x - 2cos x + 2cos³x

= 4cos³x - 3cos x

cos 3x = 4cos³x - 3cos x
___________

sin 3x = sin(2x + x) = (sin 2x)(cos x) + (cos 2x)(sin x)

= 2(sin x)(cos x)(cos x) + (1 - 2sin²x)(sin x)

= 2(sin x)(cos²x) + (sin x - 2sin³x)

= 2(sin x)(1 - sin²x) + (sin x - 2sin³x)

= 2sin x - 2sin³x + sin x - 2sin³x

= 3sin x - 4sin³x

sin 3x = 3sin x - 4sin³x

Answer 3

you're done. you have done exactly what the problem asked for, expressed the starting expression in terms of a single angle x.

Answer 4

have you tried to convert sin to cos or vice-verse

Answer 5

i have the same problem... if u get a good answer tell me