Answer to Question #139275 in Trigonometry for riz

Question #139275
1.) Isosceles triangle is similar to the Isosceles spherical triangle? justify your answer.
2.) An isosceles spherical triangle has an angle A=B= 54° and side b = 82°. Find the measure of the third angle.
3.) Determine the value of angle B of an isosceles spherical triangle ABC whose given parts are b=c= 54°28’ and a = 92°30’
4.) Solve for the side b of a right spherical triangle ABC whose parts of an isosceles spherical triangle are a = 46°, b = 75° and C = 90°.
1
Expert's answer
2020-10-21T17:22:22-0400

1) Isosceles spherical triangle - A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle.


Isosceles triangle - An isosceles triangle is a triangle that has two sides of equal length.


The above two triangle is similar as they have a common property of two sides being equal.


2) To solve an unknown angle C, convert the isosceles triangle to a right spherical triangle by constructing a "90\\degree" at the midpoint of the base.





Using Napier's rule :(for triangle ACD)


"\\sin b = \\tan x* \\tan a"


"\\cos b = 1 \/(\\tan x * \\tan a)"


"\\tan x = 1\/(\\cos b * \\tan a)"


"\\tan x = 1\/ (\\cos 82\\degree * \\tan 54\\degree)"


"\\tan x = 5.22"


so x = "79.156\\degree"


Thus solving for C:


C = 2x

= 2 * 79.156


"=" 158°18’43”



3) Let us consider the below figure the value of angle B of an isosceles spherical triangle ABC



"\\sin" co-B "= \\tan" a/2 * "\\tan" co -C


"\\cos B = \\tan a\/2 * 1\/\\tan c"


"\\cos B = \\tan 92\u00b030\u2019\/2 * 1\/\\tan 54\u00b028\u2019"


"\\therefore B = 41\u00b045\u2019"



4) Let us consider the below figure to find out the the side b of a right spherical triangle ABC



"\\sin" co-C = "\\cos a * \\cos b"


"\\cos c =" "\\cos a * \\cos b"

"\\cos b = \\cos c\/\\cos a"


"\\cos b = \\cos 75\\degree\/\\cos 46\\degree"


"\\therefore b = 68\u00b007\u2032"




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