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-   -   Can you find side length of a triangle given three angles? (http://mymathforum.com/trigonometry/343675-can-you-find-side-length-triangle-given-three-angles.html)

Larrousse March 19th, 2018 10:58 AM

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

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

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


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

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

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

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

skipjack March 19th, 2018 03:22 PM

Quote:

Originally Posted by Larrousse (Post 590261)
$\frac{\sin A}{\sin C}=a$

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

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

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 (Post 590261)
$\frac{h_c}{h_a}=a$

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

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

The above three altitude quotient equations just tell you that c = 1.

AngleWyrm2 March 19th, 2018 04:00 PM

Angles are insufficient to determine the size of a triangle.

https://s6.postimg.org/qaccs931t/Capture.png

Larrousse March 19th, 2018 04:36 PM

1 Attachment(s)
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.

Country Boy April 22nd, 2018 06:11 PM

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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