Solving Triangles

Extended Sine Rule


In triangle ABCABC, let a,ba, b and cc be the lengths of the legs of a triangle opposite vertices A,B,CA, B, C respectively. Let α,β,γ \alpha, \beta, \gamma be the values of the respective angles. What is the value of

a(sinβsinγ)+b(sinγsinα)+c(sinαsinβ)? a ( \sin \beta - \sin \gamma) + b ( \sin \gamma - \sin \alpha ) + c ( \sin \alpha - \sin \beta )?

Triangle ABCABC is inscribed in a circle Γ\Gamma. If A=60\angle A = 60^{\circ} and the length of the side opposite to vertex AA is 49349 \sqrt{3}, what is the radius of Γ\Gamma?

In triangle ABCABC, ABC=π2 \angle ABC = \frac{ \pi}{2} . If AC=12 AC = 12 , what is the radius of the circumcicle of ABC ABC ?

Triangle ABCABC has A=150\angle A=150^{\circ} and BC=7\overline{BC}=7. If the area of the circumscribed circle of ABCABC is aπa\pi, what is the value of aa?

Triangle ABCABC satisfies 2sinB=sinA+sinC2\sin B=\sin A+\sin C. If a+c=30a+c=30, what is the value of bb?

Details and assumptions

aa, bb and cc are the lengths of the sides opposite to the vertices AA, BB and CC, respectively.


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