Trig POP QUIZ #1

[ninbam1k]

Evaluate P = sin(6)sin(42)sin(66)sin(78)

in degrees btw ^

Have Fun

By hand btw, no calc ^^

[Chavi]

For x= 6°, 78°, −42°, −66°, and 30°,
we find an interesting rational angle:
sin 5x = 1/2 {e.g. 5x6 = 30,  78x5= 390 = 30}

so, sin5x = 16 (sinx)^5 − 20 (sinx)^3 + 5 sin x = 1/2
(quick expand formula)
make sinx=a
16a^5 - 20a^3 + 5a = 1/2
now (x2)  32a^5 − 40a^3 + 10a − 1 = 0
as 1/2 is a factor,
(2a − 1)(16a^4 + 8a^3 − 16a^2 − 8a + 1) = 0,

solve for the quadratic a, and then find the product of the solutions.
alternatively, vieta could do it: http://mathworld.wolfram.com/VietasFormulas.html
lol

[ninbam1k]

wow seems very complicated, i'm not sure if it's correct but ok.

PS. the answer is an exact value so yeah

[Chavi]

wow seems very complicated, i'm not sure if it's correct but ok.

PS. the answer is an exact value so yeah

what were your solutions?

[ninbam1k]

i'll PM you it, i want others to have a try

[TrueTears]

can i add an extension to this q?

change all the sin to cos, then what is the exact value?

also i ask can you immediately yield the solution from the exact value of sin(6)sin(42)sin(66)sin(78)?

If not, what is an elegant solution to find the exact value of cos(6)cos(42)cos(66)cos(78)?

[ninbam1k]

P = sin(6)sin(42)sin(66)sin(78)

insert 2cos(6) into the equation and remember to divide it out so...

  = (1 / 2cos(6))(2sin(6)cos(6)sin(42)sin(66)sin(78))

 simplify the double angle (so 2sin(6)cos(6) = sin(12))

  = (1/ 2cos(6))(sin(12)sin(42)sin(66)sin(78))

sin(78) = cos(12) complimentary angles

times 2 in, divide 2 out

  = (1/ 4cos(6))(2sin(12)cos(12)sin(42)sin(66))

sin(66) =cos(24)

times 2 in, divide 2 out

  = (1/ 8cos(6))(2sin(24)cos(24)sin(42))
  = (1/ 8cos(6))(sin(48)sin(42))

sin(42) = cos(48)

times 2 in, divide 2 out

  = (1 / 16cos(6))(2sin(48)cos(48))
  = (1 / 16cos(6))(sin(96))

sin(96) = cos(6)

  = (1 / 16cos(6))(cos(6))

cancel out the cos(6)

  = 1/16