Please could someone help or complete an unusual calculation for me. I need to solve a triangle where all the angles are known and 2 of the sides. 2 of the sides are identical and 2 of the angles are identical. One angle is one sixtieth of one degree. The other 2 angles are 180 degrees minus one sixtieth of one degree divided by 2 (just a touch under 90 degrees each) the two equal sides are 50,620,843,088,484 miles each. I would dearly like to know the length of the third side. Thank you Mike
The law of sines can't be used to get that 12 place accuracy, as you will never get that accuracy for a sine on a calculator. However, because I am so kind and generous, here are the results of a high precision calculator for the sin(PI/(60*180))
=2.908882045634245963743E-4
That ought to do it.
so let a be the side you are looking for, A the angle opposite it, b is another side, and B is the angle opposite it.
b=a*SinB/SinA
Caution, you cannot do this on an ordinary calculator, as it is limited in precision digits:(your side given has 12 digits). So you might try this one: http://keisan.casio.com/has10/Free.cgi
SinA/a=SinB/b
Looks like you are doing some astronomical type of calculation.
Make an exaggerated diagram of an isosceles triangle with a vertex angle of 1/60° or 1 minute of angle and mark each of the equal sides as 50,620,.....
I am tempted to state your side as
5.0621 x 10^13 , or just use 5.0621 in our calculation and then multiply our final answer by 10^13
Two ways:
1.
drop a perpendicular to create a right-angled triangle, let the base of that right-angled triangle be x, then its top angle would be 1/120 °
sin (1/120) = x/5.0621
x = 5.0621 sin(1/120)
= .000736..
then the base of our exaggerated triangle is
.001472505
remembering to multiply be 10^13
= 1.4725 x 10^10
2. by the cosine law
let the base be k
k^2 = 5.0621^2 + 5.0621^2 - 2(5.0621)(5.0621)cos (1/60)
k = .0014724..
times 10^13 = 1.4724 x 10^10 , same as above
On my calculator I am able to enter only 10 significant digits, so you will have to use some type of approximation.
if you have a unix system, you can use the bc calculator for arbitrary precision, as well.
Thanks bobpursley and Reiny I really appreciate the time you have obviously put into this on my behalf but I am sorry I am a very long way from understanding your explanations. In fact what I am trying to achieve is to determine how far I would have to travel from Earth in order to alter the angle of rays of light arriving from the star Sirius by 1 minute of arc. A calculation based on the creation of a circle around Sirius with Earth on its circumference yields an answer along the arc. The triangular calculation will determine the straight line distance from Earth to the same point but will be a little shorter. I have calculated the along the arc figure to be 14,730,933,173.9 by using a scientific online calculator based on distance from Sirius 8.611 lyrs. Not sure if my calculation is correct as follows
5,878,625,373,183.6 x 8.611
x2
x22
divide by 7
divide by 360
divide by 60
= 14,730,933,173.9 ???
I understand my approximation for Pi is a bit well approximate!
The real reason for wanting to solve the triangle was to confirm whether the above is correct.
I look forward to any further help.
Many thanks
Mike
Thanks also to Steve but I am sorry I am still no wiser!
Mike
compare your answer of
14,730,933,173.9
with mine: 1.4725 x 10^10
they are the same answer correct to 4 significant digits.
Had a think about the angles I described as being a touch less than 90 degrees. I have determined (I think) they are 89 and 119 over 120 degrees each
Mike
Thanks again Reiny, Thanks very much for confirming my calculation is correct although I am sitting here chuckling to myself about the way your answer is expressed. I would still love to solve the triangle if anyone will just do the calculation for me PLEASE!
Thanks
Mike
but Mike, you said that the two sides are equal, so the triangle is isosceles, and the angles would be
(180 - 1/60)/2 ° = 89.99166.. ° each
or
your top angle is 1 minute of angle
the base angles would be (10800 - 1)/2 minutes of angle
= 5399.5 minutes = 89.99166.. ° each
= 89° 59' 30''
most scientific calculators have a key labeled
D°M'S", which can be used directly to do calculations of time and angles (both in base 60)
e.g. I did
180
D°M'S"
-
0
D°M'S"
1
=
to get 89° 59' 30.00
To solve the triangle, where all the angles and two sides are known, you can use the Law of Sines or the Law of Cosines.
In this case, we can use the Law of Sines because we know the lengths of two sides and their corresponding angles. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle.
Let's assign variables for the unknown quantities:
Let the length of the third side be "c".
Given information:
- The two equal sides are 50,620,843,088,484 miles each.
- One angle is 1/60th of a degree (approximately 0.0167 degrees).
- The other two angles are both 180 degrees minus 1/60th of a degree divided by 2 (approximately 89.9917 degrees each).
Now let's apply the Law of Sines. The ratio of side lengths to their corresponding opposite angle sines is constant:
sin(A) / a = sin(B) / b = sin(C) / c
Substituting the known values:
sin(A) / 50,620,843,088,484 = sin(B) / 50,620,843,088,484 = sin(C) / c
Since angles B and C are equal, we can write:
sin(B) = sin(C), so sin(B) / 50,620,843,088,484 = sin(C) / c
Now, solving for "c":
sin(B) / 50,620,843,088,484 = sin(C) / c
sin(B) = sin(C) * 50,620,843,088,484 / c
c = sin(C) * 50,620,843,088,484 / sin(B)
Now we need to calculate the values of sin(B) and sin(C) using a calculator:
sin(B) ≈ sin(C) ≈ 0.9999999999991562
Substituting these values into the equation, we get:
c ≈ 0.9999999999991562 * 50,620,843,088,484 / 0.9999999999991562
Simplifying, we find:
c ≈ 50,620,843,088,484 miles
Therefore, the length of the third side is approximately 50,620,843,088,484 miles.