Use al-??si's method to solve a spherical triangle with sides 60^∘, 75^∘, and 31^∘

Name: Problem 39, Chapter 9: A History of Mathematics: An Introduction
Uploaded: 2020-08-25T04:35:11-08:00
Duration: 6 min 40 s
Channel: Suman Saurav Thakur
Description: Use al-??si's method to solve a spherical triangle with sides 60^∘, 75^∘, and 31^∘

Use al-??si's method to solve a spherical triangle with...

Key Concepts

Spherical Triangle

A spherical triangle is formed by the arcs of great circles on a sphere. Unlike planar triangles, its sides are measured as angles (in degrees or radians) corresponding to the angular distance along the sphere's surface. Its interior angles sum to more than 180° and it exhibits unique properties that require specialized trigonometric formulas for solution.

Spherical Law of Cosines

The spherical law of cosines provides a relationship between the sides and angles in a spherical triangle. It states that the cosine of one side is equal to the product of the cosines of the other two sides plus the product of their sines multiplied by the cosine of the included angle. This relation is essential for determining unknown sides or angles when two sides and the included angle, or three sides, are known.

Spherical Law of Sines

The spherical law of sines relates the ratios of the sine of a side to the sine of its opposite angle, and shows that these ratios are equal in any spherical triangle. This theorem is particularly useful when solving triangles with known angles and sides that are not adjacent to each other, complementing the spherical law of cosines in resolving the triangle completely.

Spherical Excess

Spherical excess is the amount by which the sum of the angles of a spherical triangle exceeds 180°. It is directly related to the area of the triangle on the sphere and is a fundamental concept in spherical geometry, illustrating the non-Euclidean nature of curved surfaces. This concept is often used in solving spherical triangles to relate geometric measurements to the sphere's surface area.

Transcript

00:01 We have 38 number problem in which we have a triangle a b c a b c and small a is equal to 614 .0 this angle a is equal to 35 degree 09 minute angle c is equal to 25 degree and 53 minutes so this is angle angle inside so case one angle angle and side okay now we will be using a plus b plus c equal to 180 degree to find angle b so angle b will be equal to 180 degree minus angle a plus angle c so let just write the values angle b will be equal to 180 degree minus angle a is 35 degree and nine minutes plus 25 degree and 53 minutes this is 180 degree minus 60 degree and 62 minutes now one thing 60 degree 60 minutes equal to 1 degree so 62 minutes will be equal to 1 degree and 2 minutes okay so this will be written as and this can also be written as 179 degree and 60 minutes minus this will become 61 degree 61 degree and 2 minutes so after subtraction we will be getting 58 minutes minutes and 9 minus 8 6 9 minus 1 8 sorry 8 7 minus 6 1 and 1 180 1 18 degree and 58 minutes so this is the value of b is the value of b now we'll be using sign law to calculate all the values of sides so from sign law sign a by a equal to sign b by b equal to sign c by see now if we take these two values we will be getting sign 35 degree or we can say just right first we will be plugging in the values later on sign a by a equal to sign b by b so from here we will be finding out b b will be equal to a into sine b by sine b by sign a now plug in the values a is 614 into sign b we have found out 118 degree 58 minutes 118 degree and 58 minutes divided by sign 35 degree and 9 minutes okay so we need to use calculator let us just find the value sign 118 degree and 58 degrees and 58 minutes divide by sign 35 degree 9 minutes 35 degree and 9 minutes 614 so this will be equal to 933 .0760 so this is the value of b now we need to find the value of c so if we take these two values again these two values so we are taking sign a by a sign a by a equal to sign c by c so c will be equal to c will be equal to a into sign c by sign a now plug in the values a means 614 into sign c means 25 degree 53 minutes divide by sign a means sign 35 degree and so let us write sign of 25 degree 53 minutes divided by a sign of 35 degree and 9 minutes into 614 so this will be 465 .667 so this is the value for c...

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