Question

sin 10 * sin 50 * sin 60 * sin70 =?

Answer 1

@satellite73 @lgbasallote @mukushla @Callisto @Hero @hartnn @experimentX

Answer 2

Hint: Calculator

Answer 3

its not that easy..

Answer 4

Yes it is. I plugged it into my calculator and got the answer

Answer 5

It got to do with some identity i forgot the name of it :/

Answer 6

@Hero no....calculator is allowed

Answer 7

Unless those stars mean something other than multiply

Answer 8

* === Multiplication.

Answer 9

Well, sin(60) is easy

Answer 10

http://www.wolframalpha.com/input/?i=sin+10+*+sin+50+*+sin+60+*+sin70+
doesn't have a nice value.

Answer 11

hm....hm...

Answer 12

sin 10 sin (60 - 10) sin(60+10)

Answer 13

sin 10 * (sin^2 60 - sin^10)

Answer 14

sin 10 * (sin^2 60 - sin^2 10)

Answer 15

i remember this, but i forgot how to do it
some addition angle formula

Answer 16

but nw wat ,,,,to do...

Answer 17

what's the point ... you are getting it all in sin 10

Answer 18

just plug the values for sin60

Answer 19

probably you should plugin the value of sin 10

Answer 20

is it \(\frac{1}{8}\)?

Answer 21

nop...

Answer 22

Yup...@satellite73 u r correct

Answer 23

@satellite73 hw u got it..???

Answer 24

memory

Answer 25

hold on .. i guess I ignored the lower part.
http://www.wolframalpha.com/input/?i=sin+10+*+sin+50+*+sin+60+*+sin70+
\[ \sqrt 3 \over 16\]

Answer 26

yep .. plugin the value of sin(10)

Answer 27

some trick of addition angle formulas will give you
\[\frac{1}{8}+\sin(10)-\sin(10)\] i think

Answer 28

Can u show the steps..

Answer 29

honestly i forget
and it if is what @experimentX wrote, then i guess i was off by \(\sin(60)\)

Answer 30

sin 10 * (sin^2 60 - sin^2 10) = sin (10) 3/4 - sin^3(10) = (sin 30)/4 = 1/8

Answer 31

@experimentX i dint get

Answer 32

Any other way....! @satellite73

Answer 33

ok give me a minute or two (or five)

Answer 34

ok...

Answer 35

oh lord here we go
we need \(\cos(x)-\cos(y)=2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})\) and lets ignore \(\sin(60)\) we know what that is, we can multiply at the end

Answer 36

take sin 60 = √3 / 2 outside

√3/2 * 2(2 sin10 * sin50 * sin 70)

Answer 37

using 2sinA sinB formula

Answer 38

√3/4 [(cos 40 - 1/2) sin 70]

Answer 39

so
\[\sin(10)\sin(50)\sin(70)=\frac{1}{2}(2\sin(10)\sin(50))\sin(70)\]
\[=\frac{1}{2}(\cos(40)-\cos(60))\sin(70)\]
\[=\frac{1}{2}(\cos(40)-\frac{1}{2})\sin(70)\]
\[=\frac{1}{2}(\cos(40)\sin(70))-\frac{1}{4}\sin(70)\]
\[=\frac{1}{4}(2\cos(40)\sin(70))-\frac{1}{4}\sin(70)\]
\[=\frac{1}{4}(\sin(110)+\sin(30))-\frac{1}{4}\sin(70)\]
\[=\frac{1}{4}(\sin(110)+\frac{1}{2})-\frac{1}{4}\sin(70)\]
this is getting exhausting

Answer 40

@satellite73 didn't you take sin 60 outside ?

Answer 41

i am ignoring it, because know what it is

Answer 42

we are almost done

Answer 43

hm..

Answer 44

@satellite73 wonderful approach......but it is ugly...)))

Answer 45

\[=\frac{1}{4}(\sin(110)+\frac{1}{2})-\frac{1}{4}\sin(70)\]
\[\frac{1}{4}\sin(110)+\frac{1}{8}-\frac{1}{4}\sin(70)\] and since \(\sin(110)=\sin(70)\) you get \(\frac{1}{8}\)

Answer 46

now you can multiply by \(\frac{\sqrt{3}}{2}\) to get the answer to your question
i am happy to see a shorter way (worked out, not with steps missing, that is not shorter)

Answer 47

but....@satellite73 the 1 st method...was much more easy..)))

Answer 48

thxxx

Answer 49

sin 10 sin 60 sin (60 - 10) sin(60+10)
sin 10 \sqrt 3/2 ( sin 60 cos 10 - cos 60 sin 10) ( sin 60 cos 10 + cos 60 sin 10)
sin 10 \sqrt 3/2 ( 3/4 cos^2 10 -1/4 sin^2 10)
sin 10 \sqrt 3/2 ( 3/4 (1 -sin^2 10) -1/4 sin^2 10)
sin 10 \sqrt 3/2 ( 3/4 -1 sin^2 10)
sin 10 \sqrt 3/2 ( 3/4 -1 sin^2 10)
\sqrt 3/2 ( 3 sin 10 -4 sin^3 10)/4
\sqrt 3/2 (sin 3*10)/4
sqrt 3/ 16 <- same as WA got
http://www.wolframalpha.com/input/?i=sin+10+*+sin+50+*+sin+60+*+sin70+