Expert: Sherman D. - 3/2/2006

Question
i have a question on half angle identities:
sin (pie)/12
cot 5(pie)/12
tan (pie)/8
and also identities where you have to prove that they are correct:
sin 3x=(sinx)(3-4sin(squared)x)
and also this:
interval (0,2pie)
sin2x=tanx

thanks so so much!

Answer
You can find this out at www.quickmath.com, click on Solve under Equations, then type in x = and your problem, then click enter and it will solve it for you but not show you how to work the problem.

sin(pi/12)
sin((pi/3) - (pi/4))
(sin(pi/3)cos(pi/4)) - (cos(pi/3)sin(pi/4))
(((sqrt(3)/2)(sqrt(2)/2)) - ((1/2)(sqrt(2)/2))
(sqrt(6)/4) - (sqrt(2)/4)
(sqrt(6) - sqrt(2))/4

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cot((5pi)/12)

cot((5pi/3) - (5pi/4))
1/(tan((5pi/3) - (5pi/4)))
1/((tan((5pi/3)) - tan((5pi)/4))/(1 + (tan(5pi/3)*tan(5pi/4)))

this is the same as saying

(1 + (tan(5pi/3)*tan(5pi/4)))/(tan(5pi/3) - tan(5pi/4))

tan(5pi/3) = -sqrt(3)
tan(5pi/4) = 1

(1 + (-sqrt(3) * 1))/(-sqrt(3) - 1)
(1 - sqrt(3))/(-1 - sqrt(3))
(1 - sqrt(3))/(-(1 + sqrt(3)))
-(1 - sqrt(3))/(1 + sqrt(3))
Multiply top and bottom by (1 - sqrt(3))
-((1 - sqrt(3))(1 - sqrt(3))/((1 + sqrt(3))(1 - sqrt(3)))
-(1 - sqrt(3) - sqrt(3) + 3)/(1 - sqrt(3) + sqrt(3) - 3)
-(4 - 2sqrt(3))/(-2)
(4 - 2sqrt(3))/2
2 - sqrt(3)

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tan(pi/8)
Since there is no way to put this in tan(x ± y) form, all i can tell you is

tan(pi/8) = .414213562

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sin(3x) = (sin(x))(3 - 4sin(x)^2)

sin(3x) = sin(x + 2x)
sin(x + 2x) = (sin(x)cos(2x)) + (cos(x)sin(2x))
sin(x+2x) = (sin(x)(1-2sin(x)^2))+(cos(x)(2sin(x)cos(x))
sin(x + 2x) = (sin(x)(1 - 2sin(x)^2) + (sin(x)(2cos(x)^2)))
sin(x + 2x) = (sin(x))(1 - 2sin(x)^2 + 2cos(x)^2)
sin(x + 2x) = (sin(x))(1 - 2sin(x)^2 + 2(1 - sin(x)^2))
sin(x + 2x) = (sin(x))(1 - 2sin(x)^2 + 2 - 2sin(x)^2)
sin(x + 2x) = (sin(x))((1 + 2) + (-2 - 2)sin(x)^2)
sin(x + 2x) = (sin(x))(3 - 4sin(x)^2)
sin(3x) = (sin(x))(3 - 4sin(x)^2)

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sin(2x) = tan(x)
sin(2x) = 2(sin(x))(cos(x))
(sin(x))/(cos(x)) = 2(sin(x))(cos(x))
(sin(x))/(sin(x)) = 2(cos(x))^2
1 = 2(cos(x))^2
(cos(x))^2 = (1/2)
cos(x) = ±(sqrt(2))/2
x = ±(pi)/4

for this

Using quickmath

Here are all the answers

x = 0, ±pi, ±(3pi)/4, ±(pi/4)

I used I used www.quickmath.com to found out how to convert it using sqrts and i used

www.math.com/tables/trig/identities.htm to find out how to work it out using formulas.