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  • 1 Post By AngleWyrm2
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March 19th, 2018, 09:58 AM   #1
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Can you find side length of a triangle given three angles?

Can you mix dimensionless function or angles to find length in triangles? For example, two sides are composed of a distance of $0.85+0.4=1.25$ and at the same time $0.4=\cos\theta$and the base is $1$?

For consecutive numbers or non-consecutive numbers $x<y<z$, I have the following example:

$(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times((1-\frac{x}{z})\times\sqrt\frac{(x+z)}{(z-x)})=\sin A$

$(\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})-(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times(\frac{x}{z})=\cos A$

$\sqrt\frac{(z-y)}{z}=\cos B$

$\sqrt\frac{y}{z}=\sin B$

$\frac{x}{z}=\cos C$

$((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin C$

$(\sqrt{\frac{y}{z}}\times\frac{x}{z})+\sqrt\frac{ z-y}{z}\times((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin A$

$(-\sqrt{\frac{z-y}{z}})\times\frac{x}{z}+\sqrt{\frac{y}{z}}\times( (1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\cos A$

The following variables $a,b,c$ represent the length of the sides of the triangles.
$\frac{\sin A}{\sin C}=a$

$\frac{\sin B}{\sin C}=b$

$\frac{\sin C}{\sin C}=c$

h=altitude
$\frac{h_c}{h_a}=a$

$\frac{h_c}{h_b}=b$

$\frac{h_c}{h_c}=c$


$((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\sin C)=\sin A$

$(\frac{\sin B}{\sin C})-((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\cos C)=\cos A$

$((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\sin C)=\sin B$

$(\frac{\sin A}{\sin C})-((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\cos C)=\cos B$

Last edited by skipjack; March 19th, 2018 at 12:04 PM.
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March 19th, 2018, 02:22 PM   #2
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Quote:
Originally Posted by Larrousse View Post
$\frac{\sin A}{\sin C}=a$

$\frac{\sin B}{\sin C}=b$

$\frac{\sin C}{\sin C}=c$
If you are given the values of the angles A, B and C, the above equations give you the values of a, b and c in terms of the angles.

If you aren't given the values of the angles A, B and C, the three equations just tell you that c = 1.

Quote:
Originally Posted by Larrousse View Post
$\frac{h_c}{h_a}=a$

$\frac{h_c}{h_b}=b$

$\frac{h_c}{h_c}=c$
The above three altitude quotient equations just tell you that c = 1.
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March 19th, 2018, 03:00 PM   #3
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Angles are insufficient to determine the size of a triangle.

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March 19th, 2018, 03:36 PM   #4
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An angle has no measurement of units, and for that reason, it is difficult to describe the length of the side of the triangle or the unit doesn't matter.
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Last edited by skipjack; March 19th, 2018 at 04:07 PM.
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April 22nd, 2018, 05:11 PM   #5
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For example, every equilateral triangle, whether its sides have length 1 cm or 1000 km, has its three angles the same with measure $\displaystyle \frac{\pi}{3}$.
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