Question

find the exact value of sin 17pi/12 using sum and difference formulas

Answer 1

Okay, first, let's review...
What is
\[\Large \sin(\pi+\alpha)\]?

Answer 2

well from what i learned sin(x+y)=sinxcosy+cosxsiny, im not quite sure what your talking about.

Answer 3

Yeah, using that formula, we do, indeed, get...
\[\Large \sin\pi\cos\alpha+\cos\pi\sin\alpha\]

Answer 4

So, what's sin π?
What's cos π?

Answer 5

okay, so what i did was split 17pi/12 into sin(8pi/12+9pi/12)=sin(4pi/3+3pi/4) but im not sure what to do from there.

Answer 6

ummm i dont know

Answer 7

Well, that works.
Now, there's a good time to apply your formula.
By the way,
sin π = 0
cos π = -1

Anyway, apply your formula for
\[\Large \sin\left(\frac{8\pi}{12}+\frac{9\pi}{12}\right)\]
Also, you might want to re-examine your reduction to lowest terms.

Answer 8

I think you'll find that
\[\Large \frac{8\pi}{12}=\frac{2\pi}{3}\ne\frac{4\pi}{3}\]

Answer 9

oh okay so if i apply every thing i got sin(2pi/3)*cos(3pi/4)+cos(2pi/3)*sin(3pi/4) right?

Answer 10

That's right :)
Have you got those values down?

Answer 11

like from a unit circle u mean?

Answer 12

From wherever, as long as you got them :)

Answer 13

But yeah, the unit circle ^.^

Answer 14

okay um 2pi/3= 120 degrees or (-1/2, sqrt3/2) and
3pi/4= 135 degrees or -(sqrt2/2, sqrt2/2)

Answer 15

Yeah... so, you got your value down?

Answer 16

okay on a calculator the answer is -(sqrt6/4) -(sqrt2/4)

Answer 17

is that right?

Answer 18

The calculator's not wrong... but did you arrive at that answer yourself?

Answer 19

nope, i dont know how to solve it out..

Answer 20

but thanks for your help anyways :)

Answer 21

@sevenupsevendown
Do you want to arrive at it manually (ie, no calculators) ?

Answer 22

yes please

Answer 23

Well, by your formula,
\[\large \sin\left(\frac{2\pi}{3}+\frac{3\pi}{4}\right)=\sin\frac{2\pi}{3}\cos\frac{3\pi}{4}+\cos\frac{2\pi}{3}\sin\frac{3\pi}{4}\]

Answer 24

ahuh i got that part then what?

Answer 25

Now, on the unit circle,
\[\huge C(\theta)=(\cos\theta,\sin\theta)\]
You got this part as well?

Answer 26

So, \[\huge C\left(\frac{2\pi}{3}\right)=C(120^o)=\left(-\frac12 \ , \ \frac{\sqrt3}2\right)\]

Answer 27

So,
\[\Large \cos\left(\frac{2\pi}{3}\right)=-\frac12\]\[\Large\sin\left(\frac{2\pi}{3}\right)=\frac{\sqrt3}2\]

Answer 28

Make your presence felt. ^.^

Answer 29

how did u get that last part?

Answer 30

\[\huge C(\theta)=(\cos\theta,\sin\theta)\]

Answer 31

Unit circle

Answer 32

oooooooooh i get it :)

Answer 33

Now, can you tell me what
\[\Large \cos\left(\frac{3\pi}{4}\right)\]and\[\Large \sin\left(\frac{3\pi}{4}\right)\]are?

Answer 34

umm the first one is -(sqrt2/2)
and the second one is (sqrt2/2)

Answer 35

Then just compute :)

Answer 36

whooo hooo! thank you!