Answer to Question #278417 in Trigonometry for Rei

Question

1. Define/describe an isosceles spherical triangle. State its properties and use illustrations if necessary.

 

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°.

 

Answer 1

An isosceles triangle definition states it as a polygon that consists of two equal sides, two equal angles, three edges, three vertices and the sum of internal angles of a triangle equal to 180⁰. In this section, we will discuss the properties of isosceles triangle along with its definitions and its significance in Maths. 

Isosceles triangle basically has two equal sides and angles opposite to these equal sides are also equal. Same like the Isosceles triangle, scalene and equilateral are also classified on the basis of their sides, whereas acute-angled, right-angled and obtuse-angled triangles are defined on the basis of angles.

An Isosceles Triangle has the following properties:

• Two sides are congruent to each other.
• The third side of an isosceles triangle which is unequal to the other two sides is called the base of the isosceles triangle.
• The two angles opposite to the equal sides are congruent to each other. That means it has two congruent base angles and this is called an isosceles triangle base angle theorem.
• The angle which is not congruent to the two congruent base angles is called an apex angle.
• The altitude from the apex of an isosceles triangle bisects the base into two equal parts and also bisects its apex angle into two equal angles.
• The altitude from the apex of an isosceles triangle divides the triangle into two congruent right-angled triangles.
• Area of Isosceles triangle = ½ × base × altitude
• Perimeter of Isosceles triangle = sum of all the three sides

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

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

$\cos B = \tan 92°30’/2 * 1/\tan 54°28’$

$\therefore B = 41°45’$

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

$\sin$ co-C = $\cos a * \cos b$

$\cos c =$ $\cos a * \cos b$

$\cos b = \cos c/\cos a$

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

$\therefore b = 68°07′$