The diameter of a circle

[Mike]

The inverse of one angle and the side opposite to the angle, which is equal to one, in all triangles gives a diameter of the circumscribed circle.

In a right triangle the inverse of the angle is always 1 which is the circle's diameter, and its' opposite's side is 1, that is hypotenuse of 1. \(\displaystyle \sin 90 =1 \) is the angle \(\displaystyle \theta \)



My question is, what is the right explanation concerning the inverse of angles with sides \(\displaystyle a,b,c \) for all \(\displaystyle \triangle ABC \) when the base or the hypotenuse equals to 1?

1)explanation

I have three sides \(\displaystyle 5, 5, 4 \) and. \(\displaystyle 1.25, 1.25, 1 \) with all the same angles except the base is 1, and all I did is divide sides \(\displaystyle 5, 5, 4\) by 4 to obtain the the simplyfied version of the triangle sides \(\displaystyle 1.25, 1.25, 1\) . From the \(\displaystyle \triangle ABC \) and 1 the side \(\displaystyle c\) opposite to the \(\displaystyle \angle C =\sqrt (1-0.68^2)\). Its' inverse is the diameter of circumscribed circle in term of \(\displaystyle \sin\) but not in terms of Cosine.

The law of Sine states that :

\(\displaystyle \frac {\sin A}{a}=\frac {\sin B}{b}=\frac {\sin C}{c}=\frac {1}{d}\)
where d is the diameter.

\(\displaystyle \frac {a}{\sin A}=\frac {b}{\sin B}=\frac {c}{\sin C}={d}\)
where d is the diameter.

 

[Dr.Peterson]

The inverse of one angle and the side opposite to the angle, which is equal to one, in all triangles gives a diameter of the circumscribed circle.

In a right triangle the inverse of the angle is always 1 which is the circle's diameter, and its' opposite's side is 1, that is hypotenuse of 1. \(\displaystyle \sin 90 =1 \) is the angle \(\displaystyle \theta \)

My question is, what is the right explanation concerning the inverse of angles with sides \(\displaystyle a,b,c \) for all \(\displaystyle \triangle ABC \) when the base or the hypotenuse equals to 1?

1)explanation

I have three sides \(\displaystyle 5, 5, 4 \) and. \(\displaystyle 1.25, 1.25, 1 \) with all the same angles except the base is 1, and all I did is divide sides \(\displaystyle 5, 5, 4\) by 4 to obtain the the simplyfied version of the triangle sides \(\displaystyle 1.25, 1.25, 1\) . From the \(\displaystyle \triangle ABC \) and 1 the side \(\displaystyle c\) opposite to the \(\displaystyle \angle C =\sqrt (1-0.68^2)\). Its' inverse is the diameter of circumscribed circle in term of \(\displaystyle \sin\) but not in terms of Cosine.

The law of Sine states that :

\(\displaystyle \frac {\sin A}{a}=\frac {\sin B}{b}=\frac {\sin C}{c}=\frac {1}{d}\)
where d is the diameter.

\(\displaystyle \frac {a}{\sin A}=\frac {b}{\sin B}=\frac {c}{\sin C}={d}\)
where d is the diameter.

Is the statement in the first paragraph translated from some other language? It makes no sense as it stands. There is no "inverse of an angle".

By comparison with the second form of the law of sines you stated, it might mean "The multiplicative inverse (reciprocal) of the sine of an angle of a triangle, times the side opposite to the angle, gives the diameter of the circumscribed circle." I'm not sure what you are saying is equal to one, or what you are asking for.

Please clarify where this comes from, and what it really means.

 

[Mike]

\(\displaystyle \frac{1}{2}\), if 2 is a diameter and the inverse of 2 =0.5, then the multiplication of \(\displaystyle \frac {1}{2} \times 2=1\) and the diameter can not equal below one because 1 is a constant and is a measure of \(\displaystyle \pi\) and angles \(\displaystyle \theta\) can not exceed 1.

 

[Dr.Peterson]

\(\displaystyle \frac{1}{2}\), if 2 is a diameter and the inverse of 2 =0.5, then the multiplication of \(\displaystyle \frac {1}{2} \times 2=1\) and the diameter can not equal below one because 1 is a constant and is a measure of \(\displaystyle \pi\) and angles \(\displaystyle \theta\) can not exceed 1.

I still don't understand. Apparently you are using "inverse" to mean "reciprocal", as I guessed; but none of the rest of this makes any sense.

Please try to state what you are asking as clearly as you can, using more words than you think necessary.

 

[Mike]

Lets say I have sums of six angles that are equal to three both Cosine and Sine.The six angles are \(\displaystyle \sqrt 0.84^2 +\sqrt 0.84^2+\sqrt(1-0.68^2)^2+0.4^2+0.4^2+0.68^2=3\).
\(\displaystyle \sqrt\frac {1}{1-0.68^2}=d\) where d is the diameter of a circle only when the base and hypotenuse equal to 1. The base c is opposite of the angle C, in a triangle A,B,C. By using the trigonometric function Sine and the law of sin I'm able to determine the diameter of the circle.