Evaluate sin18

**chengbin** · August 14th 2009, 05:44 AM

I need to calculate the exact value of sin18. How do I do that? I see that $\frac {90}{5}=18$ , so I must use some form of the addition theorems. What do I do?

BTW, how do I type the degree symbol in latex?

**mr fantastic** · August 14th 2009, 05:53 AM

Originally Posted by chengbin

I need to calculate the exact value of sin18. How do I do that? I see that $\frac {90}{5}=18$ , so I must use some form of the addition theorems. What do I do?

BTW, how do I type the degree symbol in latex?

Read this: Trigonometry--The derivation of Sin 18&#176

[tex]18^0[/tex] is easiest.

**chengbin** · August 14th 2009, 06:01 AM

Originally Posted by mr fantastic

Read this: Trigonometry--The derivation of Sin 18°

[tex]18^0[/tex] is easiest.

That's an interesting page (particularly the derivation of $\sin 1^0$ .

So if you spent enough time, you can find the exact value of all the integer degrees of $\sin$ and $\cos$ from 0 to 180?

**Amer** · August 14th 2009, 06:01 AM

Originally Posted by chengbin

I need to calculate the exact value of sin18. How do I do that? I see that $\frac {90}{5}=18$ , so I must use some form of the addition theorems. What do I do?

BTW, how do I type the degree symbol in latex?

see this link Trigonometry Angles--Pi/5 -- from Wolfram MathWorld

in the link above you can find $\cos 36^o$
but

$\cos 2x = 1-2\sin ^2 x$ so

$\cos 36^o = 1-2\sin ^2 18^o$

**Opalg** · August 14th 2009, 06:31 AM

Originally Posted by mr fantastic

[tex]18^0[/tex] is easiest.

But [tex]18^\circ[/tex] is better!

$18^0\qquad18^\circ$

**dhiab** · August 15th 2009, 12:29 AM

Originally Posted by chengbin

I need to calculate the exact value of sin18. How do I do that? I see that $\frac {90}{5}=18$ , so I must use some form of the addition theorems. What do I do?

BTW, how do I type the degree symbol in latex?

Hello This the solution :

$\begin{array}{l} let:x = 18^\circ \\ 5x = 90^\circ \\ 3x + 2x = 90^\circ \\ 3x = 90^\circ - 2x \\ \sin 3x = \cos 2x \\ \end{array}$
I'have this equation :
$\begin{array}{l} 3\sin x - 4\sin ^3 x = 1 - 2\sin ^2 x \\ let:\sin x = y \\ 4y^3 - 2y^2 - 3y + 1 = 0 \\ \left( {y - 1} \right)\left( {4y^2 + 2y - 1} \right) = 0 \\ \end{array} $
I'deduct:
$\begin{array}{l} y_1 = 1...NO \\ y_2 = \frac{{ - 1 - \sqrt 5 }}{2} < 0...NO \\ y_3 = \frac{{ - 1 + \sqrt 5 }}{2} = \sin 18^\circ \\ \end{array} $

**pacman** · August 15th 2009, 03:27 AM

sin 18 = cos 72. we have solved this already at http://www.mathhelpforum.com/math-he...tml#post344761

Working out in degrees, we have
Let x =72 degrees, multiply both sides by 5;
5x = 360
2x + 3x = 360
2x = 360-3x

Then, taking cos of both side
cos 2x = cos(360 -3x)
cos 2x = cos 3x

Using an identity for double angle and triple angle, equating them we have
2cos^2 x -1= 4cos^3 x - 3 cos x
4cos^3 x - 2cos^2 x - 3 cos x +1=0
4cos^3 x - 4cos^2 x + 2 cos^2 x - 2 cos x - cos x +1=0
4cos^2x(cos x -1) +2cos x( cos x -1) -1( cos x - 1) =0
(cos x - 1) (4cos^x +2 cos x - 1) =0

Either cos x =0 means x=90 degree
Solve (4cos^x +2 cos x - 1) =0

cos x = [-2 +(plus or minus){sq rt (4+16)] /8
cos x = [-2 +(plus or minus){sq rt (20)] /8
cos x = [-2 +2(plus or minus){sq rt (5)] /8
cos x = [-1 +(plus or minus){sq rt (5)] /4

Neglecting NEGATIVE values of cos x, because x lies in first quadrant
Therefore cos 72 degrees = (sqrt(5) -1)/4

cos 72 = 0.30901699437494742410229341718 . . . . = sin 18

**luobo** · August 15th 2009, 04:47 AM

Originally Posted by chengbin

I need to calculate the exact value of sin18. How do I do that? I see that $\frac {90}{5}=18$ , so I must use some form of the addition theorems. What do I do?

BTW, how do I type the degree symbol in latex?

Check this post.

http://www.mathhelpforum.com/math-he...tml#post350000

$\sin54^\circ-\sin18^\circ=\cos36^\circ-\sin18^\circ=1-2\sin^2 18^\circ-\sin18^\circ=\frac{1}{2}$

It is a quadratic equation.

**pacman** · August 15th 2009, 05:37 AM

luobo, nice insight. here is the graph of your equation . . . .

**pacman** · August 15th 2009, 05:39 AM

dhiab, nice solution . . . . here is the graph of your equation, 4y^3 - 2y^2 - 3y + 1 = 0.

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