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# cosine of 1/6 π r

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6 replies to this topic

### #1gasto  Members

Posted 13 March 2014 - 11:24 AM

So I want to deduce the adjacent side length of a hypotenuse at a π/6 angle  from the x axis. Basic stuff, right?

if

1/8 of a circumference is to its adjacent side

π/4

___

√(1/2)

as 1/12 of a circumference is to its adjacent side:

1/6π

____

x

Intel Core 2 Quad CPU Q6600, 2.4 GHz. 3GB RAM. ATI Radeon HD 3400.

### #2Álvaro  Members

Posted 13 March 2014 - 11:32 AM

If the hypotenuse has length 1, the adjacent side at an π/6 angle has length cos(π/6) = sqrt(3)/2. Is that what you wanted to know? Or do you want to know why it's sqrt(3)/2?

### #3gasto  Members

Posted 13 March 2014 - 11:49 AM

If the hypotenuse has length 1, the adjacent side at an π/6 angle has length cos(π/6) = sqrt(3)/2. Is that what you wanted to know? Or do you want to know why it's sqrt(3)/2?

Exactly, I haven't slept well so I am having trouble deriving it.

You can use pitagoras theorem for a π/4 angle since both sides are equal at that angle, hence:

h2 = x2 + x2

x = √(1/2)

Edited by gasto, 13 March 2014 - 11:53 AM.

Intel Core 2 Quad CPU Q6600, 2.4 GHz. 3GB RAM. ATI Radeon HD 3400.

### #4Álvaro  Members

Posted 13 March 2014 - 12:07 PM

Draw an equilateral triangle in the usual position (one horizontal side at the bottom of the figure), then draw the vertical line that passes through the top vertex. This divides the equilateral triangle in two triangles like the one you are interested in. From this construction, it's obvious that the length of the other side is 1/2. Now use the Pythagorean theorem to compute the side you are interested in.

### #5gasto  Members

Posted 14 March 2014 - 08:25 AM

I get 2(√(5/4))

Intel Core 2 Quad CPU Q6600, 2.4 GHz. 3GB RAM. ATI Radeon HD 3400.

### #6Álvaro  Members

Posted 14 March 2014 - 11:48 AM

I get 2(√(5/4))

You are doing it wrong.

Do you agree that if you split an equilateral triangle with side 1 in two by drawing a line that passes through one vertex and is perpendicular to the opposite side, the opposite side is divided in two parts of size 1/2? Now look at only one of the two triangles that we have formed. It is a right triangle with hypotenuse of length 1 and π/6 is one of its angles. The side opposite the π/6 angle has length 1/2. The remaining side has length x such that

x^2 + (1/2)^2 = 1^2

That's where you get x = sqrt(3)/2.

### #7gasto  Members

Posted 14 March 2014 - 08:06 PM

Ah basic euclidean geometry. I must be declining in intellect with these overnight refluxes... thanks to esomeprazol and sucralfate, it is improving.

Intel Core 2 Quad CPU Q6600, 2.4 GHz. 3GB RAM. ATI Radeon HD 3400.

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