Help, is this solveable?? Calculus teacher will give an automatic A if I can solve...Sin(to the)8x or Sin^8x?

Question

8
Sin x

Can this be solved?? Help!!!!

All she gave was

sin to the eight followed by x
sin8x

Rewrite Sin to the 8th x
Sin^8 x, in terms of the first power of the cosine with the use of power reducing formulas. does that make sense?

ok, my last chace, here is what i know
sin^4x = (sin^2 x)^2 property of exponents

= (1 - cos 2x)^2 over(/2) power reducing formula

= 1/4 (1 - 2 cos 2x + cos^2 2x) expand binomial

=1/4 1 - 2 cos 2x + {1 + cos4x} over ( /2 ) power reducing formula

= 1/4 -1/2 cos 2x + 1/8 +1/8 cos 4x distributive property

=3/8 - 1/2 cos 2x + 1/8 cos 4x simplify

= 1/8 (3-4 cos 2x + cos 4x) factor

SO how do i reduce

sin^8x???????????? HELP

Answer 1

Well, since sin^2(x) = 1 - cos^2(x), you could say

sin^8(x) = (1 - cos^2(x))^4 = (1 - cos(x))^4*(1 + cos(x))^4

Is that good enough?

Answer 2

If sin² x = 1 - cos² x = (1/2)(1 - cos 2x), then sin^8 x = (sin² x)^3 =
[ 1/2 (1 - cos 2x)]^3 =
1/8 ( 1 - 3 cos 2x + 3 cos² 2x - cos^3 2x) =
1/8 ( 1 - 3 cos 2x + 3 [ 1/2 (1 + cos 2x)] - cos^3 2x) =
1/8 ( 1 - 3 cos 2x + 3/2 + 1/2 cos 2x - cos^3 2x) =
1/8 - 3/8 cos 2x + 3/4 + 1/16 cos 2x - cos^3 2x =
7/8 - 5/16 cos 2x - cos^3 2x =

forget all that. I'll redo it later. (sin²)^3 = sin^6, not sin^8. I shouldn't do trig before the morning coffee. But it's typical of what you go through working out something like this. You stumble around until you get it right, and in the process you learn why it's right.

Answer 3

One technique for functions such as:
sin to the power of n (x) is by using the trig substitution cos 2x = 1 - 2 sin^2 x,
hence sin^2(x) = (1-cos 2x)/2

to integrate sin^2(x), we just integrate the right expression which is simple by using u = 2x

for higher even powers of sin x, do this repeatitively - ie. you'll get a function in terms of cos4x for sin^4 x.

Answer 4

Need more information. The sine function can operate on any real number. Sine, cosine, and tangent are the basic trigonometric functions and when subject to a power, you just multiply the result by itself so many times.
For example:
sin 30° = 0.5

(sin 30°)^2 = sin^2 30° = 0.25
(sin 30°)^3 = sin^3 30° = 0.125

It is just a matter of notation.

Answer 5

exactly the way you started but 2^3 = 8

Answer 6

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Answer 7

It most likely can be solved, otherwise she wouldn't (most likely) have given it to you asking you to solve it. Otherwise, she could have been sarcastic. I am do not know calculus otherwise I would help you, but now mor than what I have said would be a futility.

Answer 8

I agree with Philo: I don't see an equation.

See if you can write a more complete problem. There's people, like me, 'chomping at the bit' to solve a good problem.

Answer 9

Sin^8(x) or Sin^(8x)? and is the "S" supposed to be capitalized? believe it or not that matters

Answer 10

nop