Right Spherical Triangle

Right Spherical Triangle

Case of the Right spherical triangle: Right at C.
If the triangle has more than one interior angle with a value of 90^{\circ}, it is said to be oblique.

The Napier’s Circle:

Napiers circle
In the right triangle, the sides and angles are written in a consecutive way but without the right angle itself while taking the complementary angles for the quantities opposite to the right angle.

napiers1

From the graph:
\bar{B}=90^{\circ}-B
\bar{A}=90^{\circ}-A

\bar{c}=90^{\circ}-c

Sine rule for opposite parts:
The sine of any middle part is equal to the product of the cosines of its opposite parts.
Example:
\sin {a}=\cos {\bar{c}}\cos {\bar{A}}
\sin {a}=\cos {(90^{\circ}-c)}\cos {(90^{\circ}-A)}
This gives:
\sin {a}=\sin {c} \sin {A}

Sine rule for adjacent parts:
\sin {A}=\tan {b}\tan {\bar {C}}
\sin {A}=\tan {b}\tan {(90^{\circ}-C)}
\sin {A}=\tan {b}\cot {C}

We can use what we know to verify the following:
Case 1:
\sin {a}=\sin {c} \sin {A}

Case 2:
\cos {c}=\cos {b} \cos {a}

Case 3:
\cot {A}=\cos {c} \tan {B}

Case 4:
\cos {A}=\cos {a} \sin {B}

Case 5:
\tan {a}=\tan {c} \cos {B}

Case 6:
\tan {a}=\tan {A} \sin {b}

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