Question

sin 8 theta

Answer 1

What is your question(s)?

Answer 2

I ned to find a formula for sin8 theta

Answer 3

What you are asking is vague, I am still not sure what your question is.

Answer 4

\[\sin(8\theta)\]

Answer 5

using the double angle formula, i am asked to find a formula for \[\sin (8\theta)\]

Answer 6

Do you know the double-angle formula?

Answer 7

yes, \[\sin (2\theta)=2\sin \theta \cos \theta \]

Answer 8

What must you do then so that your original trigonometric function matches that of the Sine double-angle identity?

Answer 9

\[\sin 4(2\theta)\]
I don't understand beyond that.

Answer 10

Lets break down the identity

Answer 11

\[\sin(2\theta)=2\sin(\theta)\cos(\theta) \]
Do you notice what happens to the \[(2\theta)\] as it goes to the identity?

Answer 12

it gets broken into the 2 sine theta and cosine theta?

Answer 13

Yes but more specifically what happens to just the theta?

Answer 14

Hint: \[(2\theta) \] is really \[(2 \times \theta) \]

Answer 15

i have no idea. this a new concept for me so i have no clue

Answer 16

Lets look at just the sine on the left and right, what is missing from the theta after it goes into the identity?

Answer 17

the 2 next to theta

Answer 18

Right! More specifically the theta lost a multiple of 2 to become the identity on the right. So after analyzing the identity, what must \[\sin(8\theta) \] do to have a 2 next to it?

Answer 19

be \[\sin 4(2\theta)\]?

Answer 20

No for the identity we only lost a multiple of 2 not a multiple of 4 so to use that identity what must it lose?

Answer 21

a 2?

Answer 22

Correct! Awesome so what would our new identity look like on the left if it only lost a multiple of 2?

Answer 23

Hint:\[\sin(8\theta)=2\sin(?)\cos(?)\]

Answer 24

\[\sin (8\theta)=2\sin (\theta)\cos(\theta)\]

Answer 25

That is still incorrect. The multiple of only 2 was removed from the original \[(8\theta)\] so it can not be only \[2\sin(\theta)\cos(\theta)\] there must be something missing within the \[(?\theta)\] what do you think it is?

Answer 26

2

Answer 27

If we say that 2 is the missing variable and multiply the multiple of 2 back into the \[(\theta)\] it will give us \[2 \times (2\theta) = (4\theta)\] which is not our original theta so what must be the missing variable?

Answer 28

4?

Answer 29

Correct! Because we have to be consistent with our identity\[2 \sin(\theta)\cos(\theta)\] then the original function can only lose the multiple of 2 as shown which by simply algebra shows that \[2 \times ? = 8\]

Answer 30

Now that we know the missing variable is 4 by analyzing the identity what must our anwser be?

Answer 31

i have no idea what to do next